The calculator and calculus packages∗

Scientific calculations with LATEX

Robert Fuster
Universitat Politècnica de València
rfuster@mat.upv.es

2022/09/15

Abstract
The calculator package allows us to use LATEX as a calculator, with which we can perform
many of the common scientific calculations (with the limitation in accuracy imposed by the
TEX arithmetic).
This package introduces several new instructions that allow you to do several calculations
with integer and decimal numbers using LATEX. Apart from add, multiply or divide, we can
calculate powers, square roots, logarithms, trigonometric and hyperbolic functions . . .
In addition, the calculator package supports some elementary calculations with vectors
in two and three dimensions and square 2 × 2 and 3 × 3 matrices.
The calculus package adds to the calculator package several utilities to use and define
various functions and their derivatives, including elementary functions, operations with
functions, polar coordinates and vector-valued real functions.
Version 2.0 adds new capabilities to both packages. Specifically, now, calculator and
calculus can evaluate the inverse trigonometric and the inverse hyperbolic functions (so that
we can work with all the classic elementary functions), and also can do some additional
calculation with vectors (such as the cross product and the angle between two vectors).
Version 2.1 fixes some bugs and calculation problems.1

Contents

1 Introduction
The calculator package defines some instructions which allow us to realize algebraic operations
(and to evaluate elementary functions) in our documents. The operations implemented by the
calculator package include routines of assignment of variables, arithmetical calculations with
real and integer numbers, two and three dimensional vector and matrix arithmetics and the
computation of square roots, trigonometrical, exponential,
√ logarithmic and hyperbolic functions.
In addition, some important numbers, such as 2, π or e, are predefined.
∗ This document corresponds to calculator v.2.1 and calculus v.2.1, dated 2022/09/15.
1 Thanks to Schmitz Manuel, Thorsten Wolterin, Jim Cline, Schremmer Alain and July Tikhonov.

1
 The name of all these commands is spelled in capital letters (with very few exceptions: the
commands \DEGtoRAD and \RADtoDEG and the control sequences that define special numbers,
as \numberPI) and, in general, they all need one or more mandatory arguments, the first one(s)
of which is(are) number(s) and the last one(s) is(are) the name(s) of the command(s) where the
results will be stored.2 The new commands defined in this way work in any LATEX mode.
By example, this instruction
\MAX{3}{5}{\solution}

stores 5 in the command \solution. In a similar way,

\FRACTIONSIMPLIFY{10}{12}{\numerator}{\denominator}

defines \numerator and \denominator as 5 i 6, respectively.

The data arguments should not be necessarily explicit numbers; it may also consist in com-
mands the value of which is a number. This allows us to chain several calculations, since in the
following example:
Ex. 1 % \tempA=2,5^2
\SQUARE{2.5}{\tempA}
% \tempB=sqrt(12)
\SQUAREROOT{12}{\tempB}
2.52 6.25 % \tempC=exp(3,4)
√ + e3.4 = + 29.96432
12 3.4641 \EXP{3.4}{\tempC}
% \divisio=\tempA/tempB
= 1.80421 + 29.96432
\DIVIDE{\tempA}{\tempB}{\divisio}
= 31.76854 % \sol=\divisio+\tempC
\ADD{\divisio}{\tempC}{\sol}
\begin{align*}
\frac{2.5^2}{\sqrt{12}}+\mathrm{e}^{3.4}
&= \frac{\tempA}{\tempB}+\tempC \\
&= \divisio+\tempC \\
&=\sol
\end{align*}

Observe that, in this example, we have followed exactly the same steps that we would do to
2.52
calculate √ 12
+ e3.4 with a standard calculator: We would calculate the square, the root and
the exponential and, finally, we would divide and add the results.
It does not matter if the arguments results are or not predefined. But these commands act
as declarations, so that its scope is local in environments and groups.
2 Logically, the control sequences that represent special numbers (as \numberPI) does not need any argument.

2
Ex. 2 \SQUARE{5}\sol
The \texttt{\textbackslash sol}
The \sol command contains the square command contains the square of $5$:
of 5: \[5^2=\sol\]
52 = 25 \begin{center}
\SQUAREROOT{5}\sol
Now, the \sol command is the square Now, the \texttt{\textbackslash sol}
root of 5: command is the square root of $5$:
\[\sqrt{5}=\sol\]
√
5 = 2.23605 \end{center}
On having gone out of the \texttt{center}
On having gone out of the center environ- environment,
ment, the command recovers its previous the command recovers its previous value:
\sol
value: 25
The calculus package goes a step further and allows us to define and use in a user-friendly
manner various functions and their derivatives.
For exemple, using the calculus package, you can define the f (t) = t2 et − cos 2t function as
follows:
% \PRODUCTfunction{\SQUAREfunction}{\EXPfunction}{\tempfunctionA}
% \SCALEVARIABLEfunction{2}{\COSfunction}{\tempfunctionB}
% \SUBTRACTfunction{\tempfunctionA}{\tempfunctionB}{\Ffunction}
Then you cau compute any value of the new function \Ffunction and its derivative: typing
\Ffunction{hnumi}{h\sol i}{h\Dsol i}
the values of f (num) and f 0 (num) will be stored in \sol and \Dsol .

Part I
The calculator package
2 Predefined numbers
The calculator package predefines the following numbers:
\numberPI 3.14159 ≈ π \numberHALFPI 1.57079 ≈ π/2
\numberTHREEHALFPI 4.71237 ≈ 3π/2 \numberTHIRDPI 1.0472 ≈ π/3
\numberQUARTERPI 0.78539 ≈ π/4 \numberFIFTHPI 0.62831 ≈ π/5
\numberSIXTHPI 0.52359 ≈ π/6 \numberTWOPI 6.28317 ≈ 2π
\numberE 2.71828 ≈ e \numberINVE 0.36787 ≈ 1/e
\numberETWO 7.38902 ≈ e2 \numberINVETWO 0.13533 ≈ 1/e2
\numberLOGTEN 2.30258 ≈ log 10
\numberGOLD 1.61803 ≈ φ
√ \numberINVGOLD 0.61803 ≈ 1/φ
√
\numberSQRTTWO 1.41421 ≈ √2 \numberSQRTTHREE 1.73205 ≈ 3
\numberSQRTFIVE 2.23607 ≈ 5
\numberCOSXXX 0.86603 ≈ cos π/6 \numberCOSXLV 0.70711 ≈ cos π/4

3
3 Operations with numbers
3.1 Assignments and comparisons
The first command we describe here is used to store a number in a control sequence. The other
two commands in this section determine the maximum and minimum of a pair of numbers.
\COPY{hnumi}{h\cmd i} stores the number num to the command \cmd .

Ex. 3 \COPY{-1.256}{\sol}
\sol
-1.256

\MAX{hnum1 i}{hnum2 i}{h\cmd i} stores in \cmd the maximum of the numbers num1 and
num2 .

Ex. 4 \MAX{1.256}{3.214}{\sol}
\[\max(1.256,3.214)=\sol\]

max(1.256, 3.214) = 3.214

\MIN{hnum1 i}{hnum2 i}{h\cmd i} stores in \cmd the minimum of num1 and num2 .

Ex. 5 \MIN{1.256}{3.214}{\sol}
\sol
1.256

3.2 Real arithmetic

3.2.1 The four basic operations
The following commands calculate the four arithmetical basic operations.

\ADD{hnum1 i}{hnum2 i}{h\cmd i} Sum of numbers num1 and num2 .

Ex. 6 \ADD{1.256}{3.214}{\sol}
$1.256+3.214=\sol$
1.256 + 3.214 = 4.47

\SUBTRACT{hnum1 i}{hnum2 i}{h\cmd i} Difference num1 -num2 .

Ex. 7 \SUBTRACT{1.256}{3.214}{\sol}
$1.256-3.214=\sol$
1.256 − 3.214 = −1.95801

4
\MULTIPLY{hnum1 i}{hnum2 i}{h\cmd i} Product num1 ×num2 .

Ex. 8 \MULTIPLY{1.256}{3.214}{\sol}
$1.256\times3.214=\sol$
1.256 × 3.214 = 4.03677

\DIVIDE{hnum1 i}{hnum2 i}{h\cmd i} Quotient num1 /num2 .3

Ex. 9 \DIVIDE{1.256}{3.214}{\sol}
$1.256/3.214=\sol$
1.256/3.214 = 0.39078

3.2.2 Powers with integer exponent

\SQUARE{hnumi}{h\cmd i} Square of the number num .

Ex. 10 \SQUARE{-1.256}{\sol}
$(-1.256)^2=\sol$
(−1.256)2 = 1.57751

\CUBE{hnumi}{h\cmd i} Cube of num .

Ex. 11 \CUBE{-1.256}{\sol}
$(-1.256)^3=\sol$
(−1.256)3 = −1.98134

\POWER{hnumi}{hexpi}{h\cmd i} The exp power of num .

The exponent, exp , must be an integer (if you want to calculate powers with non integer
exponents, use the \EXP command).

Ex. 12 \POWER{-1.256}{-5}{\sola}
\POWER{-1.256}{5}{\solb}
\POWER{-1.256}{0}{\solc}
(−1.256)−5 = −0.31989 \[
\begin{aligned}
(−1.256)5 = −3.1256
(-1.256)^{-5}&=\sola
(−1.256)0 = 1 \\
(-1.256)^{5}&=\solb
\\
(-1.256)^{0}&=\solc
\end{aligned}
\]

3 This command uses a modified version of the division algorithm of Claudio Beccari.

5
3.2.3 Absolute value, integer part and fractional part
\ABSVALUE{hnumi}{h\cmd i} Absolute value of num .

Ex. 13 \ABSVALUE{-1.256}{\sol}
$\left\vert-1.256\right\vert=\sol$
|−1.256| = 1.256

\INTEGERPART{hnumi}{h\cmd i} Integer part of num .4

Ex. 14 \INTEGERPART{1.256}{\sola}
\INTEGERPART{-1.256}{\solb}
The integer part of 1.256 is 1, but the The integer part of $1.256$ is $\sola$,
integer part of −1.256 is −2. but the integer part of $-1.256$ is $\solb$.

\FLOOR is an alias of \INTEGERPART.

Ex. 15 \FLOOR{1.256}{\sol}
The integer part of $1.256$ is $\sol$.
The integer part of 1.256 is 1.

\FRACTIONALPART{hnumi}{h\cmd i} Fractional part of num .5

Ex. 16 \FRACTIONALPART{1.256}{\sol}
\sol
0.256
0.744 \FRACTIONALPART{-1.256}{\sol}
\sol

3.2.4 Truncation and rounding

\TRUNCATE[hni]{hnumi}{h\cmd i} truncates the number num to n decimal places.
6
\ROUND[n ]{hnumi}{h\cmd i} rounds the number num to n decimal places.
The optional argument n may be 0, 1, 2, 3 or 4 (the default is 2).7

Ex. 17 \TRUNCATE[0]{1.25688}{\sol}
\sol
1
1.25 \TRUNCATE[2]{1.25688}{\sol}
1.2568 \sol

\TRUNCATE[4]{1.25688}{\sol}
\sol
4 The integer part of x is the largest integer that is less than or equal to x.
5 code modified in version 2.1 (thanks to July Tikhonov who reported a bug and suggested the solution).
6 code modified in version 2.1 (thanks to Jim Cline and Schremmer Alain who reported a bug).
7 Note than \TRUNCATE[0] is equivalent to \INTEGERPART only for non-negative numbers.

6
 Ex. 18 \ROUND[0]{1.25688}{\sol}
\sol
1
1.26 \ROUND[2]{1.25688}{\sol}
1.2569 \sol

\ROUND[4]{1.25688}{\sol}
\sol

3.3 Integers
The operations described here are subject to the same restrictions as those referring to decimal
numbers. In particular, although TEX does not have this restriction in its integer arithmetic,
the largest integer that can be used is 16383.

3.3.1 Integer division, quotient and remainder

\INTEGERDIVISION{hnum1 i}{hnum2 i}{h\cmd1 i}{h\cmd2 i} stores in the \cmd1 and \cmd2
commands the quotient and the remainder of the integer division of the two integers num1
and num2 . The remainder is a non-negative number smaller than the divisor.8

Ex. 19 \INTEGERDIVISION{435}{27}{\sola}{\solb}
$435=27\times\sola+\solb$
435 = 27 × 16 + 3
27 = 435 × 0 + 27 \INTEGERDIVISION{27}{435}{\sola}{\solb}
−435 = 27 × (−17) + 24 $27=435\times\sola+\solb$
435 = −27 × (−16) + 3
\INTEGERDIVISION{-435}{27}{\sola}{\solb}
−435 = −27 × 17 + 24 $-435=27\times(\sola)+\solb$

\INTEGERDIVISION{435}{-27}{\sola}{\solb}
$435=-27\times(\sola)+\solb$

\INTEGERDIVISION{-435}{-27}{\sola}{\solb}
$-435=-27\times\sola+\solb$

\INTEGERQUOTIENT{hnum1 i}{hnum2 i}{h\cmd i} Integer part of the quotient of num1 and

num2 . These two numbers are not necessarily integers.
8 The scientific computing systems (such as Matlab. Scilab or Mathematica) do not always return a non-

negative residue —especially when the divisor is negative—. However, the most reasonable definition of integer
quotient is this one: the quotient of the division D/d is the largest number q for which dq ≤ D. With this
definition, the remainder r = D − qd is a non-negative number.

7
 Ex. 20 \INTEGERQUOTIENT{435}{27}{\sol}
\sol
16
0 \INTEGERQUOTIENT{27}{435}{\sol}
-17 \sol

\INTEGERQUOTIENT{-43.5}{2.7}{\sol}
\sol

\MODULO{hnum1 i}{hnum2 i}{h\cmd i} Remainder of the integer division of num1 and num2 .

Ex. 21 \MODULO{435}{27}{\sol}
\[
435 \equiv \sol \pmod{27}
435 ≡ 3 (mod 27) \]
\MODULO{-435}{27}{\sol}
−435 ≡ 24 (mod 27) \[
-435 \equiv \sol \pmod{27}
\]

3.3.2 Greatest common divisor and least common multiple

\GCD{hnum1 i}{hnum2 i}{h\cmd i} Greatest common divisor of the integers num1 and num2 .

Ex. 22 \GCD{435}{27}{\sol}
$\gcd(435,27)=\sol$
gcd(435, 27) = 3

\LCM{hnum1 i}{hnum2 i}{h\cmd i} Least common multiple of num1 and num2 .

Ex. 23 \newcommand{\lcm}{\operatorname{lcm}}
\LCM{435}{27}{\sol}
lcm(435, 27) = 3915 $\lcm(435,27)=\sol$

3.3.3 Simplifying fractions

\FRACTIONSIMPLIFY{hnum1 i}{hnum2 i}{h\cmd1 i} {h\cmd2 i} stores in the \cmd1 and \cmd2
commands the numerator and denominator of the irreducible fraction equivalent to
num1 /num2 .

Ex. 24 \FRACTIONSIMPLIFY{435}{27}{\sola}{\solb}
$435/27=\sola/\solb$
435/27 = 145/9

8
3.4 Elementary functions
3.4.1 Square roots
\SQUAREROOT {hnumi}{h\cmd i} Square root of the number num .

Ex. 25 \SQUAREROOT{1.44}{\sol}
$\sqrt{1.44}=\sol$
√
1.44 = 1.2

If the argument num is negative, the package returns a warning message.

Instead of \SQUAREROOT, you can use the alias \SQRT.

3.4.2 Exponential and logarithm

The \EXP and \LOG commands compute, by default, exponentials and logarithms of the natural
base e. They admit, however, an optional argument to choose another base.
\EXP {hnumi}{h\cmd i} Exponential of the number num .

Ex. 26 \EXP{0.5}{\sol}
$\exp(0.5)=\sol$
exp(0.5) = 1.64871

9
The argument num must be in the interval [−9.704, 9.704].
Moreover, the \EXP command accepts an optional argument, to compute expressions such
as ax :
\EXP [hnum1 i]{hnum2 i}{h\cmd i} Exponential with base num1 of num2 . num1 must be a
positive number.

Ex. 27 \EXP[10]{1.3}{\sol}
$10^{1.3}=\sol$
101.3 = 19.95209
21/3 = 1.25989 \EXP[2]{0.33333}{\sol}
$2^{1/3}=\sol$

\LOG {hnumi}{h\cmd i} logarithm of the number num .

Ex. 28 \LOG{0.5}{\sol}
$\log 0.5=\sol$
log 0.5 = −0.69315

9 9.704 is the logarithm of 16383, the largest number that supports the TEX’s arithmetic.

9
\LOG [hnum1 i]{hnum2 i}{h\cmd i} Logarithm in base num1 of num2 .

Ex. 29 \LOG[10]{0.5}{\sol}
$\log_{10} 0.5=\sol$
log10 0.5 = −0.30103

3.4.3 Trigonometric functions

The arguments, in functions \SIN, \COS, . . . , are measured in radians. If you measure angles in
degrees (sexagesimal or not), use the \DEGREESSIN, \DEGREESCOS, . . . commands.
\SIN {hnumi}{h\cmd i} Sine of num .
\COS {hnumi}{h\cmd i} Cosine of num .
\TAN {hnumi}{h\cmd i} Tangent of num .
\COT {hnumi}{h\cmd i} Cotangent of num .

Ex. 30 \SIN{\numberTHIRDPI}{\sol}
$\sin \pi/3=\sol$
sin π/3 = 0.86601
cos π/3 = 0.5 \COS{\numberTHIRDPI}{\sol}
tan π/3 = 1.73201 $\cos \pi/3=\sol$
cot π/3 = 0.57736
\TAN{\numberTHIRDPI}{\sol}
$\tan \pi/3=\sol$

\COT{\numberTHIRDPI}{\sol}
$\cot \pi/3=\sol$

\DEGREESSIN {hnumi}{h\cmd i} Sine of num sexagesimal degrees.

\DEGREESCOS {hnumi}{h\cmd i} Cosine of num sexagesimal degrees.
\DEGREESTAN {hnumi}{h\cmd i} Tangent of num sexagesimal degrees.
\DEGREESCOT {hnumi}{h\cmd i} Cotangent of num sexagesimal degrees.

Ex. 31 \DEGREESSIN{60}{\sol}
$\sin 60^{\textrm o}=\sol$
sin 60o = 0.86601
cos 60o = 0.49998 \DEGREESCOS{60}{\sol}
tan 60o = 1.73201 $\cos 60^{\textrm o}=\sol$
cot 60o = 0.57736
\DEGREESTAN{60}{\sol}
$\tan 60^{\textrm o}=\sol$

\DEGREESCOT{60}{\sol}
$\cot 60^{\textrm o}=\sol$

10
 The latter commands support an optional argument that allows us to divide the circle in an
arbitrary number of degrees (not necessarily 360).
\DEGREESSIN [hdegreesi]{hnumi}{h\cmd i}
\DEGREESCOS [hdegreesi]{hnumi}{h\cmd i}

\DEGREESTAN [hdegreesi]{hnumi}{h\cmd i}
\DEGREESCOT [hdegreesi]{hnumi}{h\cmd i}
By example, \DEGREESCOS[400]{50} computes the cosine of 50 gradians (a right angle has
100 gradians, the whole circle has 400 gradians), which are equivalent to 45 (sexagesimal) degrees
or π/4 radians. Or to 1 degree, if we divide the circle into 8 parts!
Ex. 32 \DEGREESCOS[400]{50}{\sol}
\sol
0.70709
0.70709 \DEGREESCOS{45}{\sol}
0.7071 \sol
0.70709
\COS{\numberQUARTERPI}{\sol}
\sol

\DEGREESCOS[8]{1}{\sol}
\sol

Moreover, we have a couple of commands to convert between radians and degrees,

\DEGtoRAD {hnumi}{h\cmd i} Equivalence in radians of num sexagesimal degrees.

\RADtoDEG {hnumi}{h\cmd i} Equivalence in sexagesimal degrees of num radians.

Ex. 33 \DEGtoRAD{60}{\sol}
\sol
1.0472

and two other commands to reduce arguments to basic intervals:

\REDUCERADIANSANGLE {hnumi}{h\cmd i} Reduces the arc num to the interval ] − π, π].

\REDUCEDEGREESANGLE {hnumi}{h\cmd i} Reduces the angle num to the interval ] − 180, 180].

Ex. 34 \MULTIPLY{\numberTWOPI}{10}{\TWENTYPI}
\ADD{\numberPI}{\TWENTYPI}{\TWENTYONEPI}
3.14159 \REDUCERADIANSANGLE{\TWENTYONEPI}{\sol}
90 \sol

\REDUCEDEGREESANGLE{3690}{\sol}
\sol

11
3.4.4 Hyperbolic functions
\SINH {hnumi}{h\cmd i} stores in \cmd the hyperbolic sine of num .
\COSH {hnumi}{h\cmd i} Hyperbolic cosine of num .
\TANH {hnumi}{h\cmd i} Hyperbolic tangent of num .
\COTH {hnumi}{h\cmd i} Hyperbolic cotangent of num .

Ex. 35 \SINH{1.256}{\sol}
\sol
1.61328
1.89807 \COSH{1.256}{\sol}
0.84995 \sol
1.17651
\TANH{1.256}{\sol}
\sol

\COTH{1.256}{\sol}
\sol

3.4.5 Inverse trigonometric functions(new in version 2.0)

\ARCSIN {hnumi}{h\cmd i} stores in \cmd the arcsin (inverse of sine) of num .
\ARCCOS {hnumi}{h\cmd i} arccos of num .
\ARCTAN {hnumi}{h\cmd i} arctan of num .
\ARCCOT {hnumi}{h\cmd i} arccot of num .

Ex. 36 \ARCSIN{0.5}{\sol}
\sol
0.5236
1.04718 \ARCCOS{0.5}{\sol}
1.04718 \sol
2.35619
\ARCTAN{\numberSQRTTHREE}{\sol}
\sol

\ARCCOT{-1}{\sol}
\sol

3.4.6 Inverse hyperbolic functions(new in version 2.0)

\ARSINH {hnumi}{h\cmd i} stores in \cmd the arsinh (inverse of hyperbolic sine) of num .
\ARCOSH {hnumi}{h\cmd i} arcosh of num .
\ARTANH {hnumi}{h\cmd i} artanh of num .

12
\ARCOTH {hnumi}{h\cmd i} arcoth of num .

Ex. 37 \ARSINH{1}{\sol}
\sol
0.88138
0 \ARCOSH{1}{\sol}
0.5493 \sol
0.5493
\ARTANH{0.5}{\sol}
\sol

\ARCOTH{2}{\sol}
\sol

4 Operations with lengths

\LENGTHDIVIDE{hlength1 i}{hlength2 i}{h\cmd i}
This command divides two lengths and returns a number.

Ex. 38 \LENGTHDIVIDE{1in}{1cm}{\sol}
One inch equals $\sol$ centimeters.
One inch equals 2.54 centimeters.

Commands \LENGTHADD and \LENGTHSUBTRACT return the sum and the difference of two
lengths (new in version 2.0).
\LENGTHADD{hlength1 i}{hlength2 i}{h\cmd i}
\LENGTHSUBTRACT{hlength1 i}{hlength2 i}{h\cmd i}
(\cmd must be a predefined length).

Ex. 39 \newlength{\mylength}
\LENGTHADD{1in}{1cm}{\mylength}
1in + 1cm = 100.72273pt. $1in+1cm=\the\mylength$.
1in − 1cm = 43.81725pt.
\LENGTHSUBTRACT{1in}{1cm}{\mylength}
$1in-1cm=\the\mylength$.

5 Matrix arithmetic
The calculator package defines the commands described below to operate on vectors and
matrices. We only work with two or three-dimensional vectors and 2 × 2 and 3 × 3 ma-
trices. Vectors are represented in the form (a1,a2) or (a1,a2,a3);10 and, in the case
of matrices, columns are separated à la matlab by semicolons: (a11,a12;a21,a22) or
(a11,a12,a13;a21,a22,a23;a31,a32,a33).
10 But they are column vectors.

13
5.1 Vector operations
5.1.1 Assignments
\VECTORCOPY(hx,y i)(h\cmd1,\cmd2 i) copy the entries of vector (hx,yi) to the \cmd1 and
\cmd2 commands.
\VECTORCOPY(hx,y,z i)(h\cmd1,\cmd2,\cmd3 i) copy the entries of vector (x ,y ,z ) to the
\cmd1 , \cmd2 and \cmd3 commands.

Ex. 40 \VECTORCOPY(1,-1)(\sola,\solb)
$(\sola,\solb)$
(1, −1)
(1, −1, 2) \VECTORCOPY(1,-1,2)(\sola,\solb,\solc)
$(\sola,\solb,\solc)$

5.1.2 Vector addition and subtraction

\VECTORADD(hx1,y1 i)(hx2,y2 i)(h\cmd1,\cmd2 i)

\VECTORADD(hx1,y1,z1 i)(hx2,y2,z2 i) (h\cmd1,\cmd2,\cmd3 i)

\VECTORSUB(hx1,y1 i)(hx2,y2 i)(h\cmd1,\cmd2 i)
\VECTORSUB(hx1,y1,z1 i)(hx2,y2,z2 i) (h\cmd1,\cmd2,\cmd3 i)

Ex. 41 \VECTORADD(1,-1,2)(3,5,-1)(\sola,\solb,\solc)
$(1,-1,2)+(3,5,-1)=(\sola,\solb,\solc)$
(1, −1, 2) + (3, 5, −1) = (4, 4, 1)
(1, −1, 2) − (3, 5, −1) = (−2, −6, 3) \VECTORSUB(1,-1,2)(3,5,-1)(\sola,\solb,\solc)
$(1,-1,2)-(3,5,-1)=(\sola,\solb,\solc)$

5.1.3 Scalar-vector product

\SCALARVECTORPRODUCT{hnumi}(hx,y i) (h\cmd1,\cmd2 i)

\SCALARVECTORPRODUCT{hnumi}(hx,y,z i) (h\cmd1,\cmd2,\cmd3 i)

Ex. 42 \SCALARVECTORPRODUCT{2}(3,5)(\sola,\solb)
$2(3,5)=(\sola,\solb)$
2(3, 5) = (6, 10)
2(3, 5, −1) = (6, 10, −2) \SCALARVECTORPRODUCT{2}(3,5,-1)(%
\sola,\solb,\solc)
$2(3,5,-1)=(\sola,\solb,\solc)$

14
5.1.4 Scalar (dot) product and euclidean norm
\SCALARPRODUCT(hx1,y1 i)(hx2,y2 i){h\cmd i}
\SCALARPRODUCT(hx1,y1,z1 i)(hx2,y2,z2 i){h\cmd i}
\DOTPRODUCT is an alias of \SCALARPRODUCT (new in version 2.0).
\VECTORNORM(hx,y i){h\cmd i}
\VECTORNORM(hx,y,z i){h\cmd i}

Ex. 43 \SCALARPRODUCT(1,-1)(3,5){\sol}
$(1,-1)\cdot(3,5)=\sol$
(1, −1) · (3, 5) = −2
(1, −1, 2) · (3, 5, −1) = −4 \DOTPRODUCT(1,-1,2)(3,5,-1){\sol}
k(3, 4)k = 5 $(1,-1,2)\cdot(3,5,-1)=\sol$
k(1, 2, −2)k = 3
\VECTORNORM(3,4)\sol
$\left\|(3,4)\right\|=\sol$

\VECTORNORM(1,2,-2)\sol
$\left\|(1,2,-2)\right\|=\sol$

5.1.5 Vector (cross) product (new in version 2.0)

\VECTORPRODUCT(hx1,y1,z1 i)(hx2,y2,z2 i)(h\cmd1,\cmd2,\cmd3 i)
\CROSSPRODUCT is an alias of \VECTORPRODUCT.

Ex. 44 \CROSSPRODUCT(1,-1,2)(3,5,-1)%
(\sola,\solb,\solc)
(1, −1, 2) × (3, 5, −1) = (−9, 7, 8) $(1,-1,2)\times(3,5,-1)=(\sola,\solb,\solc)$
(1, −1, 2) × (−3, 3, −6) = (0, 0, 0)
\VECTORPRODUCT(1,-1,2)(-3,3,-6)%
(\sola,\solb,\solc)
$(1,-1,2)\times(-3,3,-6)=(\sola,\solb,\solc)$

5.1.6 Unit vector parallel to a given vector (normalized vector)

\UNITVECTOR(hx,y i)(h\cmd1,\cmd2 i)
\UNITVECTOR(hx,y,z i)(h\cmd1,\cmd2,\cmd3 i)

Ex. 45 \UNITVECTOR(3,4)(\sola,\solb)
$(\sola,\solb)$
(0.59999, 0.79999)
(0.33333, 0.66666, −0.66666) \UNITVECTOR(1,2,-2)(\sola,\solb,\solc)
$(\sola,\solb,\solc)$

15
5.1.7 Absolute value (in each entry of a given vector)
\VECTORABSVALUE(hx,y i)(h\cmd1,\cmd2 i)
\VECTORABSVALUE(hx,y,z i)(h\cmd1,\cmd2,\cmd3 i)

Ex. 46 \VECTORABSVALUE(3,-4)(\sola,\solb)
$(\sola,\solb)$
(3, 4)
(3, 4, 1) \VECTORABSVALUE(3,-4,-1)(\sola,\solb,\solc)
$(\sola,\solb,\solc)$

5.1.8 Angle between two vectors (new in version 2.0)

\TWOVECTORSANGLE(hx1,y1 i)(hx2,y2 i){h\cmd i}
\TWOVECTORSANGLE(hx1,y1,z1 i)(hx2,y2,z2 i){h\cmd i}

Ex. 47 \TWOVECTORSANGLE(1,1)(0,1){\sol}
$\sol$ radians
0.78537 radians (or 44.99837 degrees) \RADtoDEG{\sol}{\degsol}
1.57079 (or 89.99937 degrees) (or $\degsol$ degrees)

\TWOVECTORSANGLE(1,0,0)(0,1,0){\sol}
$\sol$
\RADtoDEG{\sol}{\degsol}
(or $\degsol$ degrees)

5.2 Matrix operations

5.2.1 Assignments
\MATRIXCOPY (ha11,a12;a21,a22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)
 
a11 a12
Use this command to store the matrix in \cmm11 , \cmm12 , \cmm21 , \cmm22 .
a21 22
The analogous 3 × 3 version is
\MATRIXCOPY (ha11,a12,a13; [. . . ] ,a33 i) (h\cmd11,\cmd12,\cmd13; [. . . ] ,\cmd33 i)

Ex. 48 \MATRIXCOPY(1, -1, 2;

3, 0, 5;
 
1 −1 2 -1, 1, 4)%
3 0 5 (\sola,\solb,\solc;
\sold,\sole,\solf;
−1 1 4
\solg,\solh,\soli)
$\begin{bmatrix}
\sola & \solb & \solc \\
\sold & \sole & \solf \\
\solg & \solh & \soli
\end{bmatrix}$

16
 Henceforth, we will present only the syntax for commands operating with 2 × 2 matrices. In
all cases, the syntax is similar if we work with 3 × 3 matrices. In the examples, we will work
with either 2 × 2 or 3 × 3 matrices.

5.2.2 Transposed matrix

\TRANSPOSEMATRIX (ha11,a12;a21,a22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

Ex. 49 \TRANSPOSEMATRIX(1,-1;3,0)%
(\sola,\solb;\solc,\sold)
 T   $\begin{bmatrix}
1 −1 1 3
= 1 & -1 \\ 3 & 0
3 0 −1 0 \end{bmatrix}^T=\begin{bmatrix}
\sola & \solb \\ \solc & \sold
\end{bmatrix}$

5.2.3 Matrix addition and subtraction

\MATRIXADD (ha11,a12;a21,a22 i) (hb11,b12;b21,b22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

\MATRIXSUB (ha11,a12;a21,a22 i) (hb11,b12;b21,b22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

Ex. 50 \MATRIXADD(1,-1;3,0)(3,5;-3,2)%
(\sola,\solb;\solc,\sold)
     
1 −1 3 5 4 4 $\begin{bmatrix}
+ = 1 & -1 \\ 3 & 0
3 0 −3 2 0 2
      \end{bmatrix}+
1 −1 3 5 −2 −6
− = \begin{bmatrix}
3 0 −3 2 6 −2 3 & 5 \\ -3 & 2
\end{bmatrix}=\begin{bmatrix}
\sola & \solb \\ \solc & \sold
\end{bmatrix}$

\MATRIXSUB(1,-1;3,0)(3,5;-3,2)%
(\sola,\solb;\solc,\sold)
$\begin{bmatrix}
1 & -1 \\ 3 & 0
\end{bmatrix}-
\begin{bmatrix}
3 & 5 \\ -3 & 2
\end{bmatrix}=\begin{bmatrix}
\sola & \solb \\ \solc & \sold
\end{bmatrix}$

17
5.2.4 Scalar-matrix product
\SCALARMATRIXPRODUCT{hnumi} (ha11,a12;a21,a22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

Ex. 51 \SCALARMATRIXPRODUCT{3}(1,-1,2;
3, 0,5;
   
1 −1 2 3 −3 6 -1, 1,4)%
3 3 0 5 =  9 0 15 (\sola,\solb,\solc;
\sold,\sole,\solf;
−1 1 4 −3 3 12
\solg,\solh,\soli)
$3\begin{bmatrix}
1 & -1 & 2 \\ 3 & 0 & 5 \\ -1 & 1 & 4
\end{bmatrix}
=\begin{bmatrix}
\sola & \solb & \solc \\
\sold & \sole & \solf \\
\solg & \solh & \soli
\end{bmatrix}$

5.2.5 Matriu-vector product

\MATRIXVECTORPRODUCT (ha11,a12;a21,a22 i)(hx,y i) (h\cmd1,\cmd2 i)

Ex. 52 \MATRIXVECTORPRODUCT(1,-1;
0, 2)(3,5)(\sola,\solb)
    
1 −1 3 −2 $\begin{bmatrix}
= 1 & -1 \\ 0 & 2
0 2 5 10
\end{bmatrix}
\begin{bmatrix}
3 \\ 5
\end{bmatrix}
=\begin{bmatrix}
\sola \\ \solb
\end{bmatrix}$

5.2.6 Product of two square matrices

\MATRIXPRODUCT (ha11,a12;a21,a22 i) (hb11,b12;b21,b22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

18
 Ex. 53 \MATRIXPRODUCT(1,-1,2;3,0,5;-1,1,4)%
(3,5,-1;-3,2,-5;1,-2,3)%
(\sola,\solb,\solc;
\sold,\sole,\solf;
  
1 −1 2 3 5 −1 \solg,\solh,\soli)
3 0 5 −3 2 −5 \begin{multline*}
−1 1 4 1 −2 3 \begin{bmatrix}
  1 & -1 & 2 \\ 3 & 0 & 5 \\ -1 & 1 & 4
8 −1 10 \end{bmatrix}
=  14 5 12 \begin{bmatrix}
−2 −11 8 3 & 5 & -1 \\ -3 & 2 & -5 \\ 1 & -2 & 3
\end{bmatrix}\\
=\begin{bmatrix}
\sola & \solb & \solc \\
\sold & \sole & \solf \\
\solg & \solh & \soli
\end{bmatrix}
\end{multline*}

5.2.7 Determinant
\DETERMINANT (ha11,a12;a21,a22 i) {h\cmd i}

Ex. 54 \DETERMINANT(1,-1,2;3,0,5;-1,1,4){\sol}
\SpecialUsageIndex{\DETERMINANT}%
1 −1 2 $\begin{vmatrix}
3 0 5 = 18 1 & -1 & 2 \\ 3 & 0 & 5 \\ -1 & 1 & 4
\end{vmatrix}=\sol$
−1 1 4

5.2.8 Inverse matrix

\INVERSEMATRIX (ha11,a12;a21,a22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

Ex. 55 \INVERSEMATRIX(1,-1;3,5)(%
\sola,\solb;\solc,\sold)
 −1   $\begin{bmatrix}
1 −1 0.625 0.125
= 1 & -1 \\ 3 & 5
3 5 −0.375 0.125
\end{bmatrix}^{-1}=
\begin{bmatrix}
\sola & \solb \\ \solc & \sold
\end{bmatrix}$

If the given matrix is singular, the calculator package returns a warning message and the
\cmd11 , . . . , commands are marqued as undefined.

19
5.2.9 Absolute value (in each entry)
\MATRIXABSVALUE (ha11,a12;a21,a22 i) (h\cmd11,\cmd12;\cmd21,\cmd22 i)

Ex. 56 \MATRIXABSVALUE(1,-1,2;3,0,5;-1,1,4)%
(\sola,\solb,\solc;
 
1 1 2 \sold,\sole,\solf;
3 0 5 \solg,\solh,\soli)
$\begin{bmatrix}
1 1 4
\sola & \solb & \solc \\
\sold & \sole & \solf \\
\solg & \solh & \soli
\end{bmatrix}$

5.2.10 Solving a linear system

\SOLVELINEARSYSTEM
 (ha11,a12;a21,a22
     i)(hb1,b2 i)(h\cmd1,\cmd2 i) solves the linear sys-
a11 a12 x b1
tem = and stores the solution in (\cmd1 ,\cmd2 ).
a21 a22 y b2

Ex. 57 \SOLVELINEARSYSTEM(1,-1,2;3,0,5;-1,1,4)%
(-4,4,-2)%
Solving the linear system (\sola,\solb,\solc)
    Solving the linear system
1 −1 2 −4 \[
3 0 5 X =  4  \begin{bmatrix}
−1 1 4 −2 1 & -1 & 2 \\ 3 & 0 & 5 \\ -1 & 1 & 4
\end{bmatrix}\mathsf{X}=\begin{bmatrix}
3

 -4\\4\\-2
\end{bmatrix}
we obtain X =  5 
\]
−1 we obtain
$\mathsf{X}=\begin{bmatrix}
\sola \\ \solb\\ \solc
\end{bmatrix}$

If the given matrix is singular, the package calculator returns a warning message. When
system is indeterminate, in the bi-dimensional case one of the solutions is computed; if the
system is incompatible, then the \sola, . . . , commands are marqued as undefined. For
three equations systems, only determinate systems are solved.11

11 This is the only command that does not behave the same way with 2 × 2 and 3 × 3 matrices.

20
Part II
The calculus package
6 What is a function?
From the point of view of this package, a function f is a pair of formulae: the first one calculates
f (t); the other, f 0 (t). Therefore, any function is applied using three arguments: the value of the
variable t, and two command names where f (t) and f 0 (t) will be stored. For example,
\SQUAREfunction{hnumi}{h\sol i}{h\Dsol i}
computes f (t) = t2 and f 0 (t) = 2t (where t =num), and stores the results in the commands \sol
and \Dsol.12
Ex. 58 \SQUAREfunction{3}{\sol}{\Dsol}
If $f(t)=t^2$, then
If f (t) = t2 , then \[
f(3)=\sol \mbox{ and } f’(3)=\Dsol
f (3) = 9 and f 0 (3) = 6 \]

For all functions defined here, you must use the following syntax:
\functionname {hnumi}{h\cmd1 i}{h\cmd2 i}
being num a number (or a command whose value is a number), and \cmd1 and \cmd2 two
control sequence names where the values of the function and its derivative (in this number) will
be stored.
The key difference between this functions and the instructions defined in the calculator pack-
age is the inclusion of the derivative; for example, the \SQUARE{3}{\sol} instruction computes,
only, the square power of number 3, while \SQUAREfunction{3}{\sol}{\Dsol} finds, also, the
corresponding derivative.

7 Predefined functions
The calculus package predefines the most commonly used elementary functions, and includes
several utilities for defining new ones. The predefined functions are the following:
12 Do not spect any control about the existence or differentiability of the function; if the function or the

derivative are not well defined, a TEX error will occur.

21
 \ZEROfunction f (t) = 0 \ONEfunction f (t) = 1
\IDENTITYfunction f (t) = t \RECIPROCALfunction f (t) = 1/t
\SQUAREfunction f (t) = t√2 \CUBEfunction f (t) = t3
\SQRTfunction f (t) = t
\EXPfunction f (t) = exp t \LOGfunction f (t) = log t
\COSfunction f (t) = cos t \SINfunction f (t) = sin t
\TANfunction f (t) = tan t \COTfunction f (t) = cot t
\COSHfunction f (t) = cosh t \SINHfunction f (t) = sinh t
\TANHfunction f (t) = (tanh t \COTHfunction f (t) = coth t
0 si t < 0
\HEAVISIDEfunction f (t) =
1 si t ≥ 0

The following functions are added in version 2.0 (new in version 2.0)

\ARCCOSfunction f (t) = arccos t \ARCSINfunction f (t) = arcsin t

\ARCTANfunction f (t) = arctan t \ARCCOTfunction f (t) = arccot t
\ARCOSHfunction f (t) = arcosh t \ARSINHfunction f (t) = arsinh t
\ARTANHfunction f (t) = artanh t \ARCOTHfunction f (t) = arcoth t

In the following example, we use the \LOGfunction function to compute a table of the log
function and its derivative.
Ex. 59 $\begin{array}{cll}
x & \log x & \log’ x \\
x log x log0 x \LOGfunction{1}{\logx}{\Dlogx}
1 0 1 1 &\logx & \Dlogx\\
\LOGfunction{2}{\logx}{\Dlogx}
2 0.69315 0.5
2 &\logx & \Dlogx\\
3 1.0986 0.33333
\LOGfunction{3}{\logx}{\Dlogx}
4 1.38629 0.25 3 &\logx & \Dlogx\\
5 1.60942 0.2 \LOGfunction{4}{\logx}{\Dlogx}
6 1.79176 0.16666 4 &\logx & \Dlogx\\
\LOGfunction{5}{\logx}{\Dlogx}
5 &\logx & \Dlogx\\
\LOGfunction{6}{\logx}{\Dlogx}
6 &\logx & \Dlogx
\end{array}$

8 Operations with functions

We can define new functions using the following operations (the last argument is the name of
the new function):

\CONSTANTfunction{hnumi}{h\Functioni} defines \Function as the constant function num.

Example. Definition of the F (t) = 5 function:
\CONSTANTfunction{5}{\F}

22
\SUMfunction{h\function1 i}{h\function2 i} {h\Functioni} defines \Function as the sum of
functions \function1 and \function2.
Example. Definition of the F (t) = t2 + t3 function:
\SUMfunction{\SQUAREfunction}{\CUBEfunction}{\F}

\SUBTRACTfunction{h\function1 i}{h\function2 i} {h\Functioni} defines \Function as the dif-

ference of functions \function1 and \function2.
Example. Definition of the F (t) = t2 − t3 function:
\SUBTRACTfunction{\SQUAREfunction}{\CUBEfunction}{\F}
\PRODUCTfunction{h\function1 i}{h\function2 i} {h\Functioni} defines \Function as the prod-
uct of functions \function1 and \function2
Example. Definition of the F (t) = et cos t function:
\PRODUCTfunction{\EXPfunction}{\COSfunction}{\F}
\QUOTIENTfunction{h\function1 i}{h\function2 i} {h\Functioni} defines \Function as the quo-
tient of functions \function1 and \function2.
Example. Definition of the F (t) = et / cos t function:
\QUOTIENTfunction{\EXPfunction}{\COSfunction}{\F}
\COMPOSITIONfunction{h\function1 i}{h\function2 i} {h\Functioni} defines \Function as the
composition of functions \function1 and \function2.
Example. Definition of the F (t) = ecos t function:
\COMPOSITIONfunction{\EXPfunction}{\COSfunction}{\F}
(note than \COMPOSITIONfunction{f}{g}{\F} means \F= f ◦ g).
\SCALEfunction{hnumi}{h\functioni}{h\Functioni} defines \Function as the product of num-
ber num and function \function.
Example. Definition of the F (t) = 3cos t function:
\SCALEfunction{3}{\COSfunction}{\F}
\SCALEVARIABLEfunction{hnumi}{h\functioni} {h\Functioni} scales the variable by factor
num and then applies the function \function.
Example. Definition of the F (t) = cos 3t function:
\SCALEVARIABLEfunction{3}{\COSfunction}{\F}
\POWERfunction{h\functioni}{hnumi}{h\Functioni} defines \Function as the power of func-
tion \function to the exponent num (a positive integer). Example. Definition of the
F (t) = t5 function:
\POWERfunction{\IDENTITYfunction}{5}{\F}

23
\LINEARCOMBINATIONfunction{hnum1 i}{h\function1 i} {hnum2 i}{h\function2 i}{h\Functioni}
defines \Function as the linear combination of functions \function1 and \function2 mul-
tiplied, respectively, by numbers num1 and num2.
Example. Definition of the F (t) = 2t − 3 cos t function:
\LINEARCOMBINATIONfunction{2}{\IDENTITYfunction}{-3}{\COSfunction}{\F}
By combining properly this operations and the predefined functions, many elementary func-
tions can be defined.
Ex. 60 % exp(-t)
\SCALEVARIABLEfunction
If {-1}{\EXPfunction}
f (t) = 3t2 − 2e−t cos t {\NEGEXPfunction}

then % exp(-t)cos(t)
f (5) = 74.99619 \PRODUCTfunction
f 0 (5) = 29.99084 {\NEGEXPfunction}
{\COSfunction}
{\NEGEXPCOSfunction}

% 3t^2-2exp(-t)cos(t)
\LINEARCOMBINATIONfunction
{3}{\SQUAREfunction}
{-2}{\NEGEXPCOSfunction}
{\myfunction}

\myfunction{5}{\sol}{\Dsol}

If
\[
f(t)=3t^2-2\mathrm{e}^{-t}\cos t
\]
then
\[
\begin{gathered}
f(5)=\sol\\
f’(5)=\Dsol
\end{gathered}
\]

9 Polynomial functions
Although polynomial functions can be defined using linear combinations of power functions, to
facilitate our work, the calculus package includes the following commands to define more easily
the polynomials of 1, 2, and 3 degrees: \newlpoly (new linear polynomial), \newqpoly (new
quadratic polynomial), and \newcpoly (new cubic polynomial):
\newlpoly{h\Functioni}{hai}{hbi} stores the p(t) = a + b t function in the \Function com-
mand.

24
\newqpoly{h\Functioni} {hai}{hbi}{hci} stores the p(t) = a + b t + c t2 function in the
\Function command.
\newcpoly{h\Functioni}{hai}{hbi}{hci}{hd i} stores the p(t) = a + b t + c t2 + d t3 function in
the \Function command.

Ex. 61 % \mypoly=1-x^2+x^3
\newcpoly{\mypoly}{1}{0}{-1}{1}
p0 (2) = 8 \mypoly{2}{\sol}{\Dsol}
$p’(2)=\Dsol$

These declarations behave similarly to to the declaration \newcommand: If the name you want to
assign to the new function is that of an already defined command, the calculus package returns
an error message and do not redefines this command. To obtain any alternative behavior, our
package includes three other sets of declarations:

\renewlpoly, \renewqpoly, \renewcpoly redefine the already existing command \Function .

If this command does not exist, then it is not defined and an error message occurs.
\ensurelpoly, \ensureqpoly, \ensurecpoly define a new function. If the command \Function
already exists, it is not redefined.

\forcelpoly, \forceqpoly, \forcecpoly define a new function. If the command \Function

already exists, it is redefined.

10 Vector-valued functions (or parametrically defined curves)

The instruction
\PARAMETRICfunction{h\Xfunctioni}{h\Yfunctioni} {h\myvectorfunctioni}
defines the new vector-valued function f (t) = (x(t), y(t)).
The first and second arguments are a pair of functions already defined and, the third, the
name of the new function we define. Once we have defined them, the new vector functions
requires five arguments:
\myvectorfunction {hnumi}{h\cmd1 i} {h\cmd2 i}{h\cmd3 i}{h\cmd4 i}
where

• num is a number t,
• \cmd1 and \cmd2 are two command names where the values of the x(t) function and its
derivative x0 (t) will be stored, and
• \cmd3 and \cmd4 will store y(t) and y 0 (t).

25
In short, in this context, a vector function is a pair of scalar functions.
Instead of \PARAMETRICfunction we can use the alias \VECTORfunction.
Ex. 62 For the $f(t)=(t^2,t^3)$ function we have
\VECTORfunction
For the f (t) = (t2 , t3 ) function we have {\SQUAREfunction}{\CUBEfunction}{\F}

\F{4}{\solx}{\Dsolx}{\soly}{\Dsoly}
0
f (4) = (16, 64), f (4) = (8, 48)
\[
f(4)=(\solx,\soly), f’(4)=(\Dsolx,\Dsoly)
\]

11 Vector-valued functions in polar coordinates

The following instruction:
\POLARfunction{h\rfunctioni}{h\Polarfunctioni}

declares the vector function f (φ) = (r(φ) cos φ, r(φ) sin φ). The first argument is the r = r(φ)
function, (an already defined function). For example, we can define the Archimedean spiral
r(φ) = 0,5φ, as follows:
\SCALEfunction{0.5}{\IDENTITYfunction}{\rfunction}
\POLARfunction{\rfunction}{\archimedes}

12 Low-level instructions
Probably, many users of the package will not be interested in the implementation of the com-
mands this package includes. If this is your case, you can ignore this section.

12.1 The \newfunction declaration and its variants

All the functions predefined by this package use the \newfunction declaration. This control
sequence works as follows:

\newfunction{h\Functioni}{hInstructions to compute \y and \Dy from \t i}

where the second argument is the list of the instructions you need to run to calculate the value
of the function \y and the derivative \Dy in the \t point.
For example, if you want to define the f (t) = t2 + et cos t function, whose derivative is
0
f (t) = 2t + et (cos t − sin t), using the high-level instructions we defined earlier, you can write
the following instructions:
\PRODUCTfunction{\EXPfunction}{\COSfunction}{\ffunction}
\SUMfunction{\SQUAREfunction}{\ffunction}{\Ffunction}
But you can also define this function using the \newfunction command as follows:

26
 \newfunction{\Ffunction}{%
\SQUARE{\t}{\tempA} % A=t^2
\EXP{\t}{\tempB} % B=e^t
\COS{\t}{\tempC} % C=cos(t)
\SIN{\t}{\tempD} % D=sin(t)
\MULTIPLY{2}{\t}{\tempE} % E=2t
\MULTIPLY{\tempB}{\tempC}{\tempC} % C=e^t cos(t)
\MULTIPLY{\tempB}{\tempD}{\tempD} % D=e^t sin(t)
\ADD{\tempA}{\tempC}{\y} % y=t^2 + e^t cos(t)
\ADD{\tempE}{\tempC}{\tempC} % C=t^2 + e^t cos(t)
\SUBTRACT{\tempC}{\tempD}{\Dy} % y’=t^2 + e^t cos(t) - e^t sin(t)
}
It must be said, however, that the \newfunction declaration behaves similarly to \newcommand
or \newlpoly: If the name you want to assign to the new function is that of an already de-
fined command, the calculus package returns an error message and does not redefines this com-
mand. To obtain any alternative behavior, our package includes three other versions of the
\newfunction declarations: the \renewfunction, \ensurefunction and \forcefunction dec-
larations. Each of these declarations behaves differently:
\newfunction defines a new function. If the command \Function already exists, it is not
redefined and an error message occurs.
\renewfunction redefines the already existing command \Function . If this command does
not exists, then it is not defined and an error message occurs.
\ensurefunction defines a new function. If the command \Function already exists, it is not
redefined.
\forcefunction defines a new function. If the command \Function already exists, it is rede-
fined.

12.2 Vector functions and polar coordinates

You can (re)define a vector function f (t) = (x(t), y(t)) using the \newvectorfunction
declaration or any of its variants \renewvectorfunction, \ensurevectorfunction and
\forcevectorfunction:
\newvectorfunction{h\Functioni}{hInstructions to compute \x, \Dx, \y and \Dy from \t i}

For example, you can define the function f (t) = (t2 , t3 ) in the following way:

\newvectorfunction{\F}{%
\SQUARE{\t}{\x} % x=t^2
\MULTIPLY{2}{\t}{\Dx} % x’=2t
\CUBE{\t}{\y} % y=t^3
\MULTIPLY{3}{\x}{\Dy} % y’=3t^2
}

27
 Finally, to define the r = r(φ) function, in polar coordinates, we have the declarations
\newpolarfunction, \renewpolarfunction, \ensurepolarfunction and \forcepolarfunction.
\newpolarfunction{h\Functioni}{hInstructions to compute \r and \Dr from \t i}
For example, you can define the cardioide curve r(φ) = 1+cos φ, using high level instructions,

\SUMfunction{\ONEfunction}{\COSfunction}{\ffunction} % y=1 + cos t

\POLARfunction{\ffunction}{\cardioide}
or, with the \newpolarfunction declaration,
\newpolarfunction{\cardioide}{%
\COS{\t}{\r}
\ADD{1}{\r}{\r} % r=1+cos t
\SIN{\t}{\Dr}
\MULTIPLY{-1}{\Dr}{\Dr} % r’=-sin t
}

Part III
Implementation
13 calculator
1 h∗calculatori
2 \NeedsTeXFormat{LaTeX2e}
3 \ProvidesPackage{calculator}[2022/09/15 v.2.1]

13.1 Internal lengths and special numbers

\cctr@lengtha and \cctr@lengthb will be used in internal calculations and comparisons.
4 \newdimen\cctr@lengtha
5 \newdimen\cctr@lengthb

\cctr@epsilon \cctr@epsilon will store the closest to zero length in the TEX arithmetic: one scaled point
(1 sp = 1/65536 pt). This means the smallest positive number will be 0.00002 ≈ 1/65536 =
1/216 .
6 \newdimen\cctr@epsilon
7 \cctr@epsilon=1sp

\cctr@logmaxnum The largest TEX number is 16383.99998 ≈ 214 ; \cctr@logmaxnum is the logarithm of this num-
ber, 9.704 ≈ log 16384.
8 \def\cctr@logmaxnum{9.704}

28
13.2 Warning messages
9 \def\cctr@Warntruncate#1#2{%
10 \PackageWarning{calculator}%
11 {The optional argument in truncate \MessageBreak
12 must be less than 5 \MessageBreak
13 I copy #1 to #2 \MessageBreak without truncating}}
14
15 \def\cctr@Warnround#1#2{%
16 \PackageWarning{calculator}%
17 {The optional argument in round \MessageBreak
18 must be less than 5 \MessageBreak
19 I copy #1 to #2 \MessageBreak without rounding}}
20
21 \def\cctr@Warndivzero#1#2{%
22 \PackageWarning{calculator}%
23 {Division by 0.\MessageBreak
24 I can’t define #1/#2}}
25
26 \def\cctr@Warnnogcd{%
27 \PackageWarning{calculator}%
28 {gcd(0,0) is not well defined}}
29
30 \def\cctr@Warnnoposrad#1{%
31 \PackageWarning{calculator}%
32 {The argument in square root\MessageBreak
33 must be non negative\MessageBreak
34 I can’t define sqrt(#1)}}
35
36 \def\cctr@Warnnointexp#1#2{%
37 \PackageWarning{calculator}%
38 {The exponent in power function\MessageBreak
39 must be an integer\MessageBreak
40 I can’t define #1^#2}}
41
42 \def\cctr@Warnbigarcsin#1{%
43 \PackageWarning{calculator}%
44 {The argument in arcsin\MessageBreak
45 must be a number between -1 and 1\MessageBreak
46 I can’t define arcsin(#1)}}
47
48 \def\cctr@Warnbigarccos#1{%
49 \PackageWarning{calculator}%
50 {The argument in arccos\MessageBreak
51 must be a number between -1 and 1\MessageBreak
52 I can’t define arccos(#1)}}
53
54 \def\cctr@Warnsmallarcosh#1{%
55 \PackageWarning{calculator}%
56 {The argument in arcosh\MessageBreak
57 must be a number greater or equal than 1\MessageBreak

29
58 I can’t define arcosh(#1)}}
59
60 \def\cctr@Warnbigartanh#1{%
61 \PackageWarning{calculator}%
62 {The argument in artanh\MessageBreak
63 must be a number between -1 and 1\MessageBreak
64 I can’t define artanh(#1)}}
65
66 \def\cctr@Warnsmallarcoth#1{%
67 \PackageWarning{calculator}%
68 {The argument in arcoth\MessageBreak
69 must be a number greater than 1\MessageBreak
70 or smaller than -1\MessageBreak
71 I can’t define arcoth(#1)}}
72
73 \def\cctr@Warnsingmatrix#1#2#3#4{%
74 \PackageWarning{calculator}%
75 {Matrix (#1 #2 ; #3 #4) is singular\MessageBreak
76 Its inverse is not defined}}
77
78 \def\cctr@WarnsingTDmatrix#1#2#3#4#5#6#7#8#9{%
79 \PackageWarning{calculator}%
80 {Matrix (#1 #2 #3; #4 #5 #6; #7 #8 #9) is singular\MessageBreak
81 Its inverse is not defined}}
82
83 \def\cctr@WarnIncLinSys{\PackageWarning{calculator}{%
84 Incompatible linear system}}
85
86 \def\cctr@WarnIncTDLinSys{\PackageWarning{calculator}{%
87 Incompatible or indeterminate linear system\MessageBreak
88 For 3x3 systems I can solve only determinate systems}}
89
90 \def\cctr@WarnIndLinSys{\PackageWarning{calculator}{%
91 Indeterminate linear system.\MessageBreak
92 I will choose one of the infinite solutions}}
93
94 \def\cctr@WarnZeroLinSys{\PackageWarning{calculator}{%
95 0x=0 linear system. Every vector is a solution!\MessageBreak
96 I will choose the (0,0) solution}}
97
98 \def\cctr@Warninftan#1{%
99 \PackageWarning{calculator}{%
100 Undefined tangent.\MessageBreak
101 The cosine of #1 is zero and, then,\MessageBreak
102 the tangent of #1 is not defined}}
103
104 \def\cctr@Warninfcotan#1{%
105 \PackageWarning{calculator}{%
106 Undefined cotangent.\MessageBreak
107 The sine of #1 is zero and, then,\MessageBreak

30
 108 the cotangent of #1 is not defined}}
109
110 \def\cctr@Warninfexp#1{%
111 \PackageWarning{calculator}{%
112 The absolute value of the variable\MessageBreak
113 in the exponential function must be less than
114 \cctr@logmaxnum\MessageBreak
115 (the logarithm of the max number I know)\MessageBreak
116 I can’t define exp(#1)}}
117
118 \def\cctr@Warninfexpb#1#2{%
119 \PackageWarning{calculator}{%
120 The base\MessageBreak
121 in the exponential function must be positive.
122 \MessageBreak
123 I can’t define #1^(#2)}}
124
125 \def\cctr@Warninflog#1{%
126 \PackageWarning{calculator}{%
127 The value of the variable\MessageBreak
128 in the logarithm function must be positive\MessageBreak
129 I can’t define log(#1)}}
130
131 \def\cctr@Warncrossprod(#1)(#2){%
132 \PackageWarning{calculator}%
133 {Vector product only defined\MessageBreak
134 for 3 dimmensional vectors.\MessageBreak
135 I can’t define (#1)x(#2)}}
136
137 \def\cctr@Warnnoangle(#1)(#2){%
138 \PackageWarning{calculator}%
139 {Angle between two vectors only defined\MessageBreak
140 for nonzero vectors.\MessageBreak
141 I can’t define an angle between (#1) and (#2)}}

13.3 Operations with numbers

Assignements and comparisons
\COPY \COPY{h#1 i}{h#2 i} defines the #2 command as the number #1.
142 \def\COPY#1#2{\edef#2{#1}\ignorespaces}

\GLOBALCOPY Global version of \COPY. The new defined command #2 is not changed outside groups.
143 \def\GLOBALCOPY#1#2{\xdef#2{#1}\ignorespaces}

\@OUTPUTSOL \@OUTPUTSOL{h#1 i}: an internal macro to save solutions when a group is closed.
The global c.s. \cctr@outa preserves solutions. Whenever we use any temporary param-
eters in the definition of an instruction, we use a group to ensure the local character of those
parameters. The instruction \@OUTPUTSOL is a bypass to export the solution.
144 \def\@OUTPUTSOL#1{\GLOBALCOPY{#1}{\cctr@outa}\endgroup\COPY{\cctr@outa}{#1}}

31
\@OUTPUTSOLS Analogous to \@OUTPUTSOL, preserving a pair of solutions.
145 \def\@OUTPUTSOLS#1#2{\GLOBALCOPY{#1}{\cctr@outa}
146 \GLOBALCOPY{#2}{\cctr@outb}\endgroup
147 \COPY{\cctr@outa}{#1}\COPY{\cctr@outb}{#2}}

\MAX \MAX{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the maximum of numbers #1 and #2.
148 \def\MAX#1#2#3{%
149 \ifdim #1\p@ < #2\p@
150 \COPY{#2}{#3}\else\COPY{#1}{#3}\fi\ignorespaces}

\MIN \MIN{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the minimum of numbers #1 and #2.
151 \def\MIN#1#2#3{%
152 \ifdim #1\p@ > #2\p@
153 \COPY{#2}{#3}\else\COPY{#1}{#3}\fi\ignorespaces}

Real arithmetic
\ABSVALUE \ABSVALUE{h#1 i}{h#2 i} defines the #2 command as the absolute value of number #1.
154 \def\ABSVALUE#1#2{%
155 \ifdim #1\p@<\z@
156 \MULTIPLY{-1}{#1}{#2}\else\COPY{#1}{#2}\fi}

Product, sum and difference

\MULTIPLY \MULTIPLY{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the product of numbers #1 and
#2.
157 \def\MULTIPLY#1#2#3{\cctr@lengtha=#1\p@
158 \cctr@lengtha=#2\cctr@lengtha
159 \edef#3{\expandafter\strip@pt\cctr@lengtha}\ignorespaces}

\ADD \ADD{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the sum of numbers #1 and #2.
160 \def\ADD#1#2#3{\cctr@lengtha=#1\p@
161 \cctr@lengthb=#2\p@
162 \advance\cctr@lengtha by \cctr@lengthb
163 \edef#3{\expandafter\strip@pt\cctr@lengtha}\ignorespaces}

\SUBTRACT \SUBTRACT{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the difference of numbers #1 and
#2.
164 \def\SUBTRACT#1#2#3{\ADD{#1}{-#2}{#3}}

Divisions We define several kinds of divisions: the quotient of two real numbers, the integer
quotient, and the quotient of two lengths. The basic algorithm is a lightly modified version of
the Beccari’s division.
\DIVIDE \DIVIDE{h#1 i}{h#2 i}{h#3 i} defines the #3 command as the quotient of numbers #1 and
#2.
165 \def\DIVIDE#1#2#3{%
166 \begingroup

32
 Absolute values of dividend and divisor
167 \ABSVALUE{#1}{\cctr@tempD}
168 \ABSVALUE{#2}{\cctr@tempd}
The sign of quotient
169 \ifdim#1\p@<\z@\ifdim#2\p@>\z@\COPY{-1}{\cctr@sign}
170 \else\COPY{1}{\cctr@sign}\fi
171 \else\ifdim#2\p@>\z@\COPY{1}{\cctr@sign}
172 \else\COPY{-1}{\cctr@sign}\fi
173 \fi
Integer part of quotient
174 \@DIVIDE{\cctr@tempD}{\cctr@tempd}{\cctr@tempq}{\cctr@tempr}
175 \COPY{\cctr@tempq.}{\cctr@Q}
Fractional part up to five decimal places. \cctr@ndec is the number of decimal places already
computed.
176 \COPY{0}{\cctr@ndec}
177 \@whilenum \cctr@ndec<5 \do{%
Each decimal place is calculated by multiplying by 10 the last remainder and dividing it by the
divisor. But when the remainder is greater than 1638.3, an overflow occurs, because 16383.99998
is the greatest number. So, instead, we multiply the divisor by 0.1.
178 \ifdim\cctr@tempr\p@<1638\p@
179 \MULTIPLY{\cctr@tempr}{10}{\cctr@tempD}
180 \else
181 \COPY{\cctr@tempr}{\cctr@tempD}
182 \MULTIPLY{\cctr@tempd}{0.1}{\cctr@tempd}
183 \fi
184 \@DIVIDE{\cctr@tempD}{\cctr@tempd}{\cctr@tempq}{\cctr@tempr}
185 \COPY{\cctr@Q\cctr@tempq}{\cctr@Q}
186 \ADD{1}{\cctr@ndec}{\cctr@ndec}}%
Adjust the sign and return the solution.
187 \MULTIPLY{\cctr@sign}{\cctr@Q}{#3}
188 \@OUTPUTSOL{#3}}

\@DIVIDE The \@DIVIDE(h#1 i) (h#2 i)(h#3 i)(h#4 i) command computes #1/#2 and returns an integer
quotient (#3 ) and a real remainder (#4 ).
189 \def\@DIVIDE#1#2#3#4{%
190 \@INTEGERDIVIDE{#1}{#2}{#3}
191 \MULTIPLY{#2}{#3}{#4}
192 \SUBTRACT{#1}{#4}{#4}}

\@INTEGERDIVIDE \@INTEGERDIVIDE divides two numbers (not necessarily integer) and returns an integer (this is
the integer quotient only for nonnegative integers).
193 \def\@INTEGERDIVIDE#1#2#3{%
194 \cctr@lengtha=#1\p@
195 \cctr@lengthb=#2\p@
196 \ifdim\cctr@lengthb=\z@
197 \let#3\undefined

33
 198 \cctr@Warndivzero#1#2%
199 \else
200 \divide\cctr@lengtha\cctr@lengthb
201 \COPY{\number\cctr@lengtha}{#3}
202 \fi\ignorespaces}

\LENGTHADD The sum of two lengths. \LENGTHADD{h#1 i}{h#2 i}{h#3 i} stores in #3 the sum of the lenghts
#1 and #2 (#3 must be a length).
203 \def\LENGTHADD#1#2#3{\cctr@lengtha=#1
204 \cctr@lengthb=#2
205 \advance\cctr@lengtha by \cctr@lengthb
206 \setlength{#3}{\cctr@lengtha}\ignorespaces}

\LENGTHSUBTRACT The difference of two lengths. \LENGTHSUBTRACT{h#1 i}{h#2 i}{h#3 i} stores in #3 the differ-
ence of the lenghts #1 and #2 (#3 must be a length).
207 \def\LENGTHSUBTRACT#1#2#3{%
208 \LENGTHADD{#1}{-#2}{#3}}

\LENGTHDIVIDE The quotient of two lengths must be a number (not a length). For example, one inch over one
centimeter equals 2.54. \LENGTHDIVIDE{h#1 i}{h#2 i}{h#3 i} stores in #3 the quotient of the
lenghts #1 and #2.
209 \def\LENGTHDIVIDE#1#2#3{%
210 \begingroup
211 \cctr@lengtha=#1
212 \cctr@lengthb=#2
213 \edef\cctr@tempa{\expandafter\strip@pt\cctr@lengtha}%
214 \edef\cctr@tempb{\expandafter\strip@pt\cctr@lengthb}%
215 \DIVIDE{\cctr@tempa}{\cctr@tempb}{#3}
216 \@OUTPUTSOL{#3}}

Powers
\SQUARE \SQUARE{h#1 i}{h#2 i} stores #1 squared in #2.
217 \def\SQUARE#1#2{\MULTIPLY{#1}{#1}{#2}}

\CUBE \CUBE{h#1 i}{h#2 i} stores #1 cubed in #2.

218 \def\CUBE#1#2{\MULTIPLY{#1}{#1}{#2}\MULTIPLY{#2}{#1}{#2}}

\POWER \POWER{h#1 i}{h#2 i}{h#3 i} stores in #3 the power #1#2

219 \def\POWER#1#2#3{%
220 \begingroup
221 \INTEGERPART{#2}{\cctr@tempexp}
222 \ifdim \cctr@tempexp\p@<#2\p@
223 \cctr@Warnnointexp{#1}{#2}
224 \let#3\undefined
225 \else
This ensures that power will be defined only if the exponent is an integer.
226 \@POWER{#1}{#2}{#3}\fi\@OUTPUTSOL{#3}}

34
 227 \def\@POWER#1#2#3{%
228 \begingroup
229 \ifdim #2\p@<\z@
For negative exponents, an = (1/a)−n .
230 \DIVIDE{1}{#1}{\cctr@tempb}
231 \MULTIPLY{-1}{#2}{\cctr@tempc}
232 \@POWER{\cctr@tempb}{\cctr@tempc}{#3}
233 \else
234 \COPY{0}{\cctr@tempa}
235 \COPY{1}{#3}
236 \@whilenum \cctr@tempa<#2 \do {%
237 \MULTIPLY{#1}{#3}{#3}
238 \ADD{1}{\cctr@tempa}{\cctr@tempa}}%
239 \fi\@OUTPUTSOL{#3}}

Integer arithmetic and related things

\INTEGERDIVISION \INTEGERDIVISION{h#1 i}{h#2 i}{h#3 i}{h#4 i} computes the division #1/#2 and returns an
integer quotient and a positive remainder.
240 \def\INTEGERDIVISION#1#2#3#4{%
241 \begingroup
242 \ABSVALUE{#2}{\cctr@tempd}
243 \@DIVIDE{#1}{#2}{#3}{#4}
244 \ifdim #4\p@<\z@
245 \ifdim #1\p@<\z@
246 \ifdim #2\p@<\z@
247 \ADD{#3}{1}{#3}
248 \else
249 \SUBTRACT{#3}{1}{#3}
250 \fi
251 \ADD{#4}{\cctr@tempd}{#4}
252 \fi\fi\@OUTPUTSOLS{#3}{#4}}

\MODULO \MODULO{h#1 i}{h#2 i}{h#3 i} returns the remainder of division #1/#2.

253 \def\MODULO#1#2#3{%
254 \begingroup
255 \INTEGERDIVISION{#1}{#2}{\cctr@temp}{#3}\@OUTPUTSOL{#3}}

\INTEGERQUOTIENT \INTEGERQUOTIENT{h#1 i}{h#2 i}{h#3 i} returns the integer quotient of division #1/#2.
256 \def\INTEGERQUOTIENT#1#2#3{%
257 \begingroup
258 \INTEGERDIVISION{#1}{#2}{#3}{\cctr@temp}\@OUTPUTSOL{#3}}

\INTEGERPART \INTEGERPART{h#1 i}{h#2 i} returns the integer part of #2.

259 \def\@@INTEGERPART#1.#2.#3)#4{\ifnum #11=1 \COPY{0}{#4}
260 \else \ADD{0}{#1}{#4}\fi}
261 \def\@INTEGERPART#1#2{\expandafter\@@INTEGERPART#1..){#2}}
262 \def\INTEGERPART#1#2{\begingroup
263 \ifdim #1\p@<\z@

35
 264 \MULTIPLY{-1}{#1}{\cctr@temp}
265 \INTEGERPART{\cctr@temp}{#2}
266 \ifdim #2\p@<\cctr@temp\p@
267 \SUBTRACT{-#2}{1}{#2}
268 \else \COPY{-#2}{#2}
269 \fi
270 \else
271 \@INTEGERPART{#1}{#2}
272 \fi\@OUTPUTSOL{#2}}

\FLOOR \FLOOR is an alias for \INTEGERPART.

273 \let\FLOOR\INTEGERPART

\FRACTIONALPART \FRACTIONALPART{h#1 i}{h#2 i} returns the fractional part of #2.

274 \def\@@FRACTIONALPART#1.#2.#3)#4{\ifnum #21=1 \COPY{0}{#4}
275 \else \ADD{0}{0.#2}{#4}\fi}
276 \def\@FRACTIONALPART#1#2{\expandafter\@@FRACTIONALPART#1..){#2}}
277 \def\FRACTIONALPART#1#2{\begingroup
278 \ifdim #1\p@<\z@
279 \INTEGERPART{#1}{\cctr@tempA}
280 \SUBTRACT{#1}{\cctr@tempA}{#2}
281 \else
282 \@FRACTIONALPART{#1}{#2}
283 \fi\@OUTPUTSOL{#2}}
284

\TRUNCATE \TRUNCATE[h#1 i]{h#2 i}{h#3 i} truncates #2 to #1 (0, 1, 2 (default), 3 or 4) digits.

285 \def\TRUNCATE{\@ifnextchar[\@@TRUNCATE\@TRUNCATE}
286 \def\@TRUNCATE#1#2{\@@TRUNCATE[2]{#1}{#2}}
287 \def\@@TRUNCATE[#1]#2#3{%
288 \begingroup
289 \ifdim #1\p@ > 4\p@ \cctr@Warntruncate{#2}{\noexpand#3} \COPY{#2}{#3}
290 \else
291 \INTEGERPART{#2}{\cctr@tempa}
292 \ifdim \cctr@tempa\p@ = #2\p@
293 \expandafter\@@@TRUNCATE\cctr@tempa.00000.)[#1]{#3}
294 \else
295 \expandafter\@@@TRUNCATE#200000.)[#1]{#3}
296 \fi\fi
297 \@OUTPUTSOL{#3}}
298
299 \def\@@@TRUNCATE#1.#2#3#4#5#6.#7)[#8]#9{%
300 \ifcase #8
301 \COPY{#1}{#9}
302 \or\COPY{#1.#2}{#9}
303 \or\COPY{#1.#2#3}{#9}
304 \or\COPY{#1.#2#3#4}{#9}
305 \or\COPY{#1.#2#3#4#5}{#9}
306 \fi}

36
\ROUND \ROUND[h#1 i]{h#2 i}{h#3 i} rounds #2 to #1 (0, 1, 2 (default), 3 or 4) digits.
307 \def\ROUND{\@ifnextchar[\@@ROUND\@ROUND}
308 \def\@ROUND#1#2{\@@ROUND[2]{#1}{#2}}
309 \def\@@ROUND[#1]#2#3{%
310 \begingroup
311 \ifdim #1\p@ > 4\p@ \cctr@Warnround{#2}{\noexpand#3} \COPY{#2}{#3}
312 \else
313 \INTEGERPART{#2}{\cctr@tempa}
314 \ifdim \cctr@tempa\p@ = #2\p@
315 \expandafter\@@@ROUND\cctr@tempa.00000.)[#1]{#3}
316 \else
317 \expandafter\@@@ROUND#200000.)[#1]{#3}
318 \fi
319 \fi
320 \@OUTPUTSOL{#3}}
321
322 \def\@@@ROUND#1.#2#3#4#5#6.#7)[#8]#9{%
323 \ifcase #8
324 \COPY{#1}{#9} \ifnum #2>4 \ADD{#1}{1}{\cctr@tempp}\COPY{\cctr@tempp}{#9} \fi
325 \or\COPY{#1.#2}{#9} \ifnum #3>4 \ADD{#2}{1}\cctr@tempq\COPY{#1}{\cctr@tempp}
326 \ifnum\cctr@tempq=10\ADD{\cctr@tempp}{1}\cctr@tempp\COPY{0}{\cctr
327 \COPY{\cctr@tempp.\cctr@tempq}{#9}
328 \fi
329 \or\COPY{#1.#2#3}{#9} \ifnum #4>4 \ADD{#3}1\cctr@tempq\COPY{#2}{\cctr@tempp}\COPY{#1}{\cctr@tempo
330 \ifnum\cctr@tempq=10\ADD{\cctr@tempp}{1}{\cctr@tempp}\COPY{0}{\cc
331 \ifnum\cctr@tempp=10\ADD{\cctr@tempo}{1}\cctr@tempo\COPY{0}{\cctr
332 \COPY{\cctr@tempo.\cctr@tempp\cctr@tempq}{#9}
333 \fi
334 \or\COPY{#1.#2#3#4}{#9} \ifnum #5>4 \ADD{#4}1\cctr@tempq\COPY{#3}{\cctr@tempp}\COPY{#2}{\cctr@tem
335 \ifnum\cctr@tempq=10\ADD{\cctr@tempp}{1}{\cctr@tempp}\COPY{0}{\cc
336 \ifnum\cctr@tempp=10\ADD{\cctr@tempo}{1}\cctr@tempo\COPY{0}{\cctr
337 \ifnum\cctr@tempo=10\ADD{\cctr@tempn}{1}\cctr@tempn\COPY{0}{\cctr
338 \COPY{\cctr@tempn.\cctr@tempo\cctr@tempp\cctr@tempq}{#9}
339 \fi
340 \or\COPY{#1.#2#3#4#5}{#9} \ifnum #6>4 \ADD{#5}1\cctr@tempq\COPY{#4}{\cctr@tempp}\COPY{#3}{\cctr@
341 \ifnum\cctr@tempq=10\ADD{\cctr@tempp}{1}{\cctr@tempp}\COPY{0}{\cc
342 \ifnum\cctr@tempp=10\ADD{\cctr@tempo}{1}\cctr@tempo\COPY{0}{\cctr
343 \ifnum\cctr@tempo=10\ADD{\cctr@tempn}{1}\cctr@tempn\COPY{0}{\cctr
344 \ifnum\cctr@tempn=10\ADD{\cctr@tempm}{1}\cctr@tempm\COPY{0}{\cctr
345 \COPY{\cctr@tempm.\cctr@tempn\cctr@tempo\cctr@tempp\cctr@tempq}{#
346 \fi}

\GCD \GCD{h#1 i}{h#2 i}{h#3 i} Greatest common divisor, using the Euclidean algorithm
347 \def\GCD#1#2#3{%
348 \begingroup
349 \ABSVALUE{#1}{\cctr@tempa}
350 \ABSVALUE{#2}{\cctr@tempb}
351 \MAX{\cctr@tempa}{\cctr@tempb}{\cctr@tempc}
352 \MIN{\cctr@tempa}{\cctr@tempb}{\cctr@tempa}
353 \COPY{\cctr@tempc}{\cctr@tempb}

37
 354 \ifnum \cctr@tempa = 0
355 \ifnum \cctr@tempb = 0
356 \cctr@Warnnogcd
357 \let#3\undefined
358 \else
359 \COPY{\cctr@tempb}{#3}
360 \fi
361 \else
Euclidean algorithm: if c ≡ b (mod a) then gcd(b, a) = gcd(a, c). Iterating this property, we
obtain gcd(b, a) as the last nonzero residual.
362 \@whilenum \cctr@tempa > \z@ \do {%
363 \COPY{\cctr@tempa}{#3}%
364 \MODULO{\cctr@tempb}{\cctr@tempa}{\cctr@tempc}%
365 \COPY\cctr@tempa\cctr@tempb%
366 \COPY\cctr@tempc\cctr@tempa}
367 \fi\ignorespaces\@OUTPUTSOL{#3}}

\LCM \LCM{h#1 i}{h#2 i}{h#3 i} Least common multiple.

368 \def\LCM#1#2#3{%
369 \GCD{#1}{#2}{#3}%
370 \ifx #3\undefined \COPY{0}{#3}
371 \else
372 \DIVIDE{#1}{#3}{#3}
373 \MULTIPLY{#2}{#3}{#3}
374 \ABSVALUE{#3}{#3}
375 \fi}

\FRACTIONSIMPLIFY \FRACTIONSIMPLIFY{h#1 i}{h#2 i}{h#3 i}{h#4 i} Fraction simplification: #3/#4 is the irre-
ducible fraction equivalent to #1/#2.
376 \def\FRACTIONSIMPLIFY#1#2#3#4{%
377 \ifnum #1=\z@
378 \COPY{0}{#3}\COPY{1}{#4}
379 \else
380 \GCD{#1}{#2}{#3}%
381 \DIVIDE{#2}{#3}{#4}
382 \DIVIDE{#1}{#3}{#3}
383 \ifnum #4<0 \MULTIPLY{-1}{#4}{#4}\MULTIPLY{-1}{#3}{#3}\fi
384 \fi\ignorespaces}

Elementary functions
Square roots
\SQUAREROOT \SQUAREROOT{h#1 i}{h#2 i} defines #2 as the square root of #1, using the Newton’s method:
xn+1 = xn − (x2n − #1)/(2xn ).
385 \def\SQUAREROOT#1#2{%
386 \begingroup
387 \ifdim #1\p@ = \z@
388 \COPY{0}{#2}

38
 389 \else
390 \ifdim #1\p@ < \z@
391 \let#2\undefined
392 \cctr@Warnnoposrad{#1}%
393 \else
We take #1 as the initial approximation.
394 \COPY{#1}{#2}
\cctr@lengthb will be the difference of two successive iterations.
We start with \cctr@lengthb=5\p@ to ensure almost one iteration.
395 \cctr@lengthb=5\p@
Successive iterations
396 \@whilenum \cctr@lengthb>\cctr@epsilon \do {%
Copy the actual approximation to \cctr@tempw
397 \COPY{#2}{\cctr@tempw}
398 \DIVIDE{#1}{\cctr@tempw}{\cctr@tempz}
399 \ADD{\cctr@tempw}{\cctr@tempz}{\cctr@tempz}
400 \DIVIDE{\cctr@tempz}{2}{\cctr@tempz}
Now, \cctr@tempz is the new approximation.
401 \COPY{\cctr@tempz}{#2}
Finally, we store in \cctr@lengthb the difference of the two last approximations, finishing the
loop.
402 \SUBTRACT{#2}{\cctr@tempw}{\cctr@tempw}
403 \cctr@lengthb=\cctr@tempw\p@%
404 \ifnum
405 \cctr@lengthb<\z@ \cctr@lengthb=-\cctr@lengthb
406 \fi}
407 \fi\fi\@OUTPUTSOL{#2}}

\SQRT \SQRT is an alias for \SQUAREROOT.

408 \let\SQRT\SQUAREROOT

Trigonometric functions For a variable close enough to zero, the sine and tangent functions
are computed using some continued fractions. Then, all trigonometric functions are derived
from well-known formulas.
\SIN \SIN{h#1 i}{h#2 i}. Sine of #1.
409 \def\SIN#1#2{%
410 \begingroup
Exact sine for t ∈ {π/2, −π/2, 3π/2}
411 \ifdim #1\p@=-\numberHALFPI\p@ \COPY{-1}{#2}
412 \else
413 \ifdim #1\p@=\numberHALFPI\p@ \COPY{1}{#2}
414 \else
415 \ifdim #1\p@=\numberTHREEHALFPI\p@ \COPY{-1}{#2}
416 \else

39
 If |t| > π/2, change t to a smaller value.
417 \ifdim#1\p@<-\numberHALFPI\p@
418 \ADD{#1}{\numberTWOPI}{\cctr@tempb}
419 \SIN{\cctr@tempb}{#2}
420 \else
421 \ifdim #1\p@<\numberHALFPI\p@
Compute the sine.
422 \@BASICSINE{#1}{#2}
423 \else
424 \ifdim #1\p@<\numberTHREEHALFPI\p@
425 \SUBTRACT{\numberPI}{#1}{\cctr@tempb}
426 \SIN{\cctr@tempb}{#2}
427 \else
428 \SUBTRACT{#1}{\numberTWOPI}{\cctr@tempb}
429 \SIN{\cctr@tempb}{#2}
430 \fi\fi\fi\fi\fi\fi\@OUTPUTSOL{#2}}

\@BASICSINE \@BASICSINE{h#1 i}{h#2 i} applies this approximation:

x
sin x =
x2
1+
2 · 3x2
2 · 3 − x2 +
4 · 5x2
4 · 5 − x2 +
6 · 7 − x2 + · · ·
431 \def\@BASICSINE#1#2{%
432 \begingroup
433 \ABSVALUE{#1}{\cctr@tempa}
Exact sine of zero
434 \ifdim\cctr@tempa\p@=\z@ \COPY{0}{#2}
435 \else
For t very close to zero, sin t ≈ t.
436 \ifdim \cctr@tempa\p@<0.009\p@\COPY{#1}{#2}
437 \else
Compute the continued fraction.
438 \SQUARE{#1}{\cctr@tempa}
439 \DIVIDE{\cctr@tempa}{42}{#2}
440 \SUBTRACT{1}{#2}{#2}
441 \MULTIPLY{#2}{\cctr@tempa}{#2}
442 \DIVIDE{#2}{20}{#2}
443 \SUBTRACT{1}{#2}{#2}
444 \MULTIPLY{#2}{\cctr@tempa}{#2}
445 \DIVIDE{#2}{6}{#2}
446 \SUBTRACT{1}{#2}{#2}
447 \MULTIPLY{#2}{#1}{#2}
448 \fi\fi\@OUTPUTSOL{#2}}

40
 \COS \COS{h#1 i}{h#2 i}. Cosine of #1 : cos t = sin(t + π/2).
449 \def\COS#1#2{%
450 \begingroup
451 \ADD{\numberHALFPI}{#1}{\cctr@tempc}
452 \SIN{\cctr@tempc}{#2}\@OUTPUTSOL{#2}}

\TAN \TAN{h#1 i}{h#2 i}. Tangent of #1.

453 \def\TAN#1#2{%
454 \begingroup
Tangent is infinite for t = ±π/2
455 \ifdim #1\p@=-\numberHALFPI\p@
456 \cctr@Warninftan{#1}
457 \let#2\undefined
458 \else
459 \ifdim #1\p@=\numberHALFPI\p@
460 \cctr@Warninftan{#1}
461 \let#2\undefined
462 \else
If |t| > π/2, change t to a smaller value.
463 \ifdim #1\p@<-\numberHALFPI\p@
464 \ADD{#1}{\numberPI}{\cctr@tempb}
465 \TAN{\cctr@tempb}{#2}
466 \else
467 \ifdim #1\p@<\numberHALFPI\p@
Compute the tangent.
468 \@BASICTAN{#1}{#2}
469 \else
470 \SUBTRACT{#1}{\numberPI}{\cctr@tempb}
471 \TAN{\cctr@tempb}{#2}
472 \fi\fi\fi\fi\@OUTPUTSOL{#2}}

\@BASICTAN \@BASICTAN{h#1 i}{h#2 i} applies this approximation:

1
tan x =
1 1
−
x 3 1
−
x 5 1
−
x 7 1
−
x 9 1
−
x 11
− ···
x
473 \def\@BASICTAN#1#2{%
474 \begingroup
475 \ABSVALUE{#1}{\cctr@tempa}
Exact tangent of zero.
476 \ifdim\cctr@tempa\p@=\z@ \COPY{0}{#2}
477 \else

41
 For t very close to zero, tan t ≈ t.
478 \ifdim\cctr@tempa\p@<0.04\p@
479 \COPY{#1}{#2}
480 \else
Compute the continued fraction.
481 \DIVIDE{#1}{11}{#2}
482 \DIVIDE{9}{#1}{\cctr@tempa}
483 \SUBTRACT{\cctr@tempa}{#2}{#2}
484 \DIVIDE{1}{#2}{#2}
485 \DIVIDE{7}{#1}{\cctr@tempa}
486 \SUBTRACT{\cctr@tempa}{#2}{#2}
487 \DIVIDE{1}{#2}{#2}
488 \DIVIDE{5}{#1}{\cctr@tempa}
489 \SUBTRACT{\cctr@tempa}{#2}{#2}
490 \DIVIDE{1}{#2}{#2}
491 \DIVIDE{3}{#1}{\cctr@tempa}
492 \SUBTRACT{\cctr@tempa}{#2}{#2}
493 \DIVIDE{1}{#2}{#2}
494 \DIVIDE{1}{#1}{\cctr@tempa}
495 \SUBTRACT{\cctr@tempa}{#2}{#2}
496 \DIVIDE{1}{#2}{#2}
497 \fi\fi\@OUTPUTSOL{#2}}

\COT \COT{h#1 i}{h#2 i}. Cotangent of #1 : If cos t = 0 then cot t = 0; if tan t = 0 then cot t = ∞.
Otherwise, cot t = 1/ tan t.
498 \def\COT#1#2{%
499 \begingroup
500 \COS{#1}{#2}
501 \ifdim #2\p@ = \z@
502 \COPY{0}{#2}
503 \else
504 \TAN{#1}{#2}
505 \ifdim #2\p@ = \z@
506 \cctr@Warninfcotan{#1}
507 \let#2\undefined
508 \else
509 \DIVIDE{1}{#2}{#2}
510 \fi\fi\@OUTPUTSOL{#2}}

\DEGtoRAD \DEGtoRAD{h#1 i}{h#2 i}. Convert degrees to radians.

511 \def\DEGtoRAD#1#2{\DIVIDE{#1}{57.29578}{#2}}

\RADtoDEG \RADtoDEG{h#1 i}{h#2 i}. Convert radians to degrees.

512 \def\RADtoDEG#1#2{\MULTIPLY{#1}{57.29578}{#2}}

\REDUCERADIANSANGLE Reduces to the trigonometrically equivalent arc in ]−π, π].

513 \def\REDUCERADIANSANGLE#1#2{%
514 \COPY{#1}{#2}
515 \ifdim #1\p@ < -\numberPI\p@

42
 516 \ADD{#1}{\numberTWOPI}{#2}
517 \REDUCERADIANSANGLE{#2}{#2}
518 \fi
519 \ifdim #1\p@ > \numberPI\p@
520 \SUBTRACT{#1}{\numberTWOPI}{#2}
521 \REDUCERADIANSANGLE{#2}{#2}
522 \fi
523 \ifdim #1\p@ = -180\p@ \COPY{\numberPI}{#2} \fi}

\REDUCEDEGREESANGLE Reduces to the trigonometrically equivalent angle in ]−180, 180].

524 \def\REDUCEDEGREESANGLE#1#2{%
525 \COPY{#1}{#2}
526 \ifdim #1\p@ < -180\p@
527 \ADD{#1}{360}{#2}
528 \REDUCEDEGREESANGLE{#2}{#2}
529 \fi
530 \ifdim #1\p@ > 180\p@
531 \SUBTRACT{#1}{360}{#2}
532 \REDUCEDEGREESANGLE{#2}{#2}
533 \fi
534 \ifdim #1\p@ = -180\p@ \COPY{180}{#2} \fi}

Trigonometric functions in degrees Four next commands compute trigonometric func-

tions in degrees. By default, a circle has 360 degrees, but we can use an arbitrary number of
divisions using the optional argument of these commands.
\DEGREESSIN \DEGREESSIN[h#1 i]{h#2 i}{h#3 i}. Sine of #2 degrees.
535 \def\DEGREESSIN{\@ifnextchar[\@@DEGREESSIN\@DEGREESSIN}

\DEGREESCOS \DEGREESCOS[h#1 i]{h#2 i}{h#3 i}. Cosine of #2 degrees.

536 \def\DEGREESCOS{\@ifnextchar[\@@DEGREESCOS\@DEGREESCOS}

\DEGREESTAN \DEGREESTAN[h#1 i]{h#2 i}{h#3 i}. Tangent of #2 degrees.

537 \def\DEGREESTAN{\@ifnextchar[\@@DEGREESTAN\@DEGREESTAN}

\DEGREESCOT \DEGREESCOT[h#1 i]{h#2 i}{h#3 i}. Cotangent of #2 degrees.

538 \def\DEGREESCOT{\@ifnextchar[\@@DEGREESCOT\@DEGREESCOT}

\@DEGREESSIN \@DEGREESSIN computes the sine in sexagesimal degrees.

539 \def\@DEGREESSIN#1#2{%
540 \begingroup
541 \ifdim #1\p@=-90\p@ \COPY{-1}{#2}
542 \else
543 \ifdim #1\p@=90\p@ \COPY{1}{#2}
544 \else
545 \ifdim #1\p@=270\p@ \COPY{-1}{#2}
546 \else
547 \ifdim#1\p@<-90\p@
548 \ADD{#1}{360}{\cctr@tempb}

43
 549 \DEGREESSIN{\cctr@tempb}{#2}
550 \else
551 \ifdim #1\p@<90\p@
552 \DEGtoRAD{#1}{\cctr@tempb}
553 \@BASICSINE{\cctr@tempb}{#2}
554 \else
555 \ifdim #1\p@<270\p@
556 \SUBTRACT{180}{#1}{\cctr@tempb}
557 \DEGREESSIN{\cctr@tempb}{#2}
558 \else
559 \SUBTRACT{#1}{360}{\cctr@tempb}
560 \DEGREESSIN{\cctr@tempb}{#2}
561 \fi\fi\fi\fi\fi\fi\@OUTPUTSOL{#2}}

\@DEGREESCOS \@DEGREESCOS computes the cosine in sexagesimal degrees.

562 \def\@DEGREESCOS#1#2{%
563 \begingroup
564 \ADD{90}{#1}{\cctr@tempc}
565 \DEGREESSIN{\cctr@tempc}{#2}\@OUTPUTSOL{#2}}

\@DEGREESTAN \@DEGREESTAN computes the tangent in sexagesimal degrees.

566 \def\@DEGREESTAN#1#2{%
567 \begingroup
568 \ifdim #1\p@=-90\p@
569 \cctr@Warninftan{#1}
570 \let#2\undefined
571 \else
572 \ifdim #1\p@=90\p@
573 \cctr@Warninftan{#1}
574 \let#2\undefined
575 \else
576 \ifdim #1\p@<-90\p@
577 \ADD{#1}{180}{\cctr@tempb} \DEGREESTAN{\cctr@tempb}{#2}
578 \else
579 \ifdim #1\p@<90\p@
580 \DEGtoRAD{#1}{\cctr@tempb}
581 \@BASICTAN{\cctr@tempb}{#2}
582 \else
583 \SUBTRACT{#1}{180}{\cctr@tempb}
584 \DEGREESTAN{\cctr@tempb}{#2}
585 \fi\fi\fi\fi\@OUTPUTSOL{#2}}

\@DEGREESCOT \@DEGREESCOT computes the cotangent in sexagesimal degrees.

586 \def\@DEGREESCOT#1#2{%
587 \begingroup
588 \DEGREESCOS{#1}{#2}
589 \ifdim #2\p@ = \z@
590 \COPY{0}{#2}
591 \else
592 \DEGREESTAN{#1}{#2}

44
 593 \ifdim #2\p@ = \z@
594 \cctr@Warninfcotan{#1}
595 \let#2\undefined
596 \else
597 \DIVIDE{1}{#2}{#2}
598 \fi\fi\@OUTPUTSOL{#2}}

For an arbitrary number of degrees, we normalise to 360 degrees and, then, call the former
functions.
\@@DEGREESSIN \@@DEGREESSIN computes the sine. A circle has #1 degrees.
599 \def\@@DEGREESSIN[#1]#2#3{\@CONVERTDEG{#1}{#2}
600 \@DEGREESSIN{\@DEGREES}{#3}}

\@@DEGREESCOS \@@DEGREESCOS computes the sine. A circle has #1 degrees.

601 \def\@@DEGREESCOS[#1]#2#3{\@CONVERTDEG{#1}{#2}
602 \DEGREESCOS{\@DEGREES}{#3}}

\@@DEGREESTAN \@@DEGREESTAN computes the sine. A circle has #1 degrees.

603 \def\@@DEGREESTAN[#1]#2#3{\@CONVERTDEG{#1}{#2}
604 \DEGREESTAN{\@DEGREES}{#3}}

\@@DEGREESCOT \@@DEGREESCOT computes the sine. A circle has #1 degrees.

605 \def\@@DEGREESCOT[#1]#2#3{\@CONVERTDEG{#1}{#2}
606 \DEGREESCOT{\@DEGREES}{#3}}

\@CONVERTDEG \@CONVERTDEG normalises to sexagesimal degrees.

607 \def\@CONVERTDEG#1#2{\DIVIDE{#2}{#1}{\@DEGREES}
608 \MULTIPLY{\@DEGREES}{360}{\@DEGREES}}

Exponential functions
\EXP \EXP[h#1 i]{h#2 i}{h#3 i} computes the exponential #3 = #1#2 . Default for #1 is number
e.
609 \def\EXP{\@ifnextchar[\@@EXP\@EXP}

\@@EXP \@@EXP[h#1 i]{h#2 i}{h#3 i} computes #3 = #1#2

610 \def\@@EXP[#1]#2#3{%
611 \begingroup
#1 must be a positive number.
612 \ifdim #1\p@<\cctr@epsilon
613 \cctr@Warninfexpb{#1}{#2}
614 \let#3\undefined
615 \else
ab = exp(b log a).
616 \LOG{#1}{\cctr@log}
617 \MULTIPLY{#2}{\cctr@log}{\cctr@log}
618 \@EXP{\cctr@log}{#3}
619 \fi\@OUTPUTSOL{#3}}

45
 \@EXP \@EXP{h#1 i}{h#2 i} computes #3 = e#2
620 \def\@EXP#1#2{%
621 \begingroup
622 \ABSVALUE{#1}{\cctr@absval}
If |t| is greater than \cctr@logmaxnum then exp t is too large.
623 \ifdim \cctr@absval\p@>\cctr@logmaxnum\p@
624 \cctr@Warninfexp{#1}
625 \let#2\undefined
626 \else
627 \ifdim #1\p@ < \z@
2
We call \@BASICEXP when t ∈ [−6, 3]. Otherwise we use the equality exp t = (exp t/2) .
628 \ifdim #1\p@ > -6.00002\p@
629 \@BASICEXP{#1}{#2}
630 \else
631 \DIVIDE{#1}{2}{\cctr@expt}
632 \@EXP{\cctr@expt}{\cctr@expy}
633 \SQUARE{\cctr@expy}{#2}
634 \fi
635 \else
636 \ifdim #1\p@ < 3.00002\p@
637 \@BASICEXP{#1}{#2}
638 \else
639 \DIVIDE{#1}{2}{\cctr@expt}
640 \@EXP{\cctr@expt}{\cctr@expy}
641 \SQUARE{\cctr@expy}{#2}
642 \fi
643 \fi\fi\@OUTPUTSOL{#2}}

\@BASICEXP \@BASICEXP{h#1 i}{h#2 i} applies this approximation:

2x
exp x ≈ 1 +
x2 /6
2−x+
x2 /60
1+
x2 /140
1+
x2 /256
1+
x2
1+
396
644 \def\@BASICEXP#1#2{%
645 \begingroup
646 \SQUARE{#1}\cctr@tempa
647 \DIVIDE{\cctr@tempa}{396}{#2}
648 \ADD{1}{#2}{#2}
649 \DIVIDE\cctr@tempa{#2}{#2}
650 \DIVIDE{#2}{256}{#2}
651 \ADD{1}{#2}{#2}
652 \DIVIDE\cctr@tempa{#2}{#2}
653 \DIVIDE{#2}{140}{#2}

46
 654 \ADD{1}{#2}{#2}
655 \DIVIDE\cctr@tempa{#2}{#2}
656 \DIVIDE{#2}{60}{#2}
657 \ADD{1}{#2}{#2}
658 \DIVIDE\cctr@tempa{#2}{#2}
659 \DIVIDE{#2}{6}{#2}
660 \ADD{2}{#2}{#2}
661 \SUBTRACT{#2}{#1}{#2}
662 \DIVIDE{#1}{#2}{#2}
663 \MULTIPLY{2}{#2}{#2}
664 \ADD{1}{#2}{#2}\@OUTPUTSOL{#2}}

Hyperbolic functions
\COSH \COSH. Hyperbolic cosine: cosh t = (exp t + exp(−t))/2.
665 \def\COSH#1#2{%
666 \begingroup
667 \ABSVALUE{#1}{\cctr@absval}
668 \ifdim \cctr@absval\p@>\cctr@logmaxnum\p@
669 \cctr@Warninfexp{#1}
670 \let#2\undefined
671 \else
672 \EXP{#1}{\cctr@expx}
673 \MULTIPLY{-1}{#1}{\cctr@minust}
674 \EXP{\cctr@minust}{\cctr@expminusx}
675 \ADD{\cctr@expx}{\cctr@expminusx}{#2}
676 \DIVIDE{#2}{2}{#2}
677 \fi\@OUTPUTSOL{#2}}

\SINH \SINH. Hyperbolic sine: sinh t = (exp t − exp(−t))/2.

678 \def\SINH#1#2{%
679 \begingroup
680 \ABSVALUE{#1}{\cctr@absval}
681 \ifdim \cctr@absval\p@>\cctr@logmaxnum\p@
682 \cctr@Warninfexp{#1}
683 \let#2\undefined
684 \else
685 \EXP{#1}{\cctr@expx}
686 \MULTIPLY{-1}{#1}{\cctr@minust}
687 \EXP{\cctr@minust}{\cctr@expminusx}
688 \SUBTRACT{\cctr@expx}{\cctr@expminusx}{#2}
689 \DIVIDE{#2}{2}{#2}
690 \fi\@OUTPUTSOL{#2}}

\TANH \TANH. Hyperbolic tangent: tanh t = sinh t/cosh t.

691 \def\TANH#1#2{%
692 \begingroup
693 \ABSVALUE{#1}{\cctr@absval}
694 \ifdim \cctr@absval\p@>\cctr@logmaxnum\p@
695 \cctr@Warninfexp{#1}

47
 696 \let#2\undefined
697 \else
698 \SINH{#1}{\cctr@tanhnum}
699 \COSH{#1}{\cctr@tanhden}
700 \DIVIDE{\cctr@tanhnum}{\cctr@tanhden}{#2}
701 \fi\@OUTPUTSOL{#2}}

\COTH \COTH. Hyperbolic cotangent coth t = cosh t/sinh t.

702 \def\COTH#1#2{%
703 \begingroup
704 \ABSVALUE{#1}{\cctr@absval}
705 \ifdim \cctr@absval\p@>\cctr@logmaxnum\p@
706 \cctr@Warninfexp{#1}
707 \let#2\undefined
708 \else
709 \SINH{#1}{\cctr@tanhden}
710 \COSH{#1}{\cctr@tanhnum}
711 \DIVIDE\cctr@tanhnum\cctr@tanhden{#2}
712 \fi\@OUTPUTSOL{#2}}

Logarithm
\LOG \LOG[h#1 i]{h#2 i}{h#3 i} computes the logarithm #3 = log#1 #2. Default for #1 is number
e.
713 \def\LOG{\@ifnextchar[\@@LOG\@LOG}

\@LOG \@LOG{h#1 i}{h#2 i} computes #2 = log #1

714 \def\@LOG#1#2{%
715 \begingroup
The argument t must be positive.
716 \ifdim #1\p@<\cctr@epsilon
717 \cctr@Warninflog{#1}
718 \let#2\undefined
719 \else
720 \ifdim #1\p@ > \numberETWO\p@
If t > e2 , log t = log e + log(t/e) = 1 + log(t/e)
721 \DIVIDE{#1}{\numberE}{\cctr@ae}
722 \@LOG{\cctr@ae}{#2}
723 \ADD{1}{#2}{#2}
724 \else
725 \ifdim #1\p@ < 1\p@
If t < 1, log t = log(1/e) + log(te) = −1 + log(te)
726 \MULTIPLY{\numberE}{#1}{\cctr@ae}
727 \LOG{\cctr@ae}{#2}
728 \SUBTRACT{#2}{1}{#2}
729 \else

48
 For t ∈ [1, e2 ] we call \@@BASICLOG.
730 \@BASICLOG{#1}{#2}
731 \fi\fi\fi\@OUTPUTSOL{#2}}

\@@LOG \@@LOG[h#1 i]{h#2 i}{h#3 i} computes #3 = log#1 #2 = log(#2)/ log(#1)

732 \def\@@LOG[#1]#2#3{\begingroup
733 \@LOG{#1}{\cctr@loga}
734 \@LOG{#2}{\cctr@logx}
735 \DIVIDE{\cctr@logx}{\cctr@loga}{#3}\@OUTPUTSOL{#3}}

\@BASICLOG \@BASICLOG{h#1 i}{h#2 i} applies the Newton’s method to calculate x = log t:

t
xn+1 = xn + −1
exn
736 \def\@BASICLOG#1#2{\begingroup
737 % We take $\textit{\#1}-1$ as the initial approximation.
738 % \begin{macrocode}
739 \SUBTRACT{#1}{1}{\cctr@tempw}
We start with \cctr@lengthb=5\p@ to ensure almost one iteration.
740 \cctr@lengthb=5\p@%
741 \cctr@epsilon=2\cctr@epsilon%
Successive iterations
742 \@whilenum \cctr@lengthb>\cctr@epsilon \do {%
743 \COPY{\cctr@tempw}{\cctr@tempoldw}
744 \EXP{\cctr@tempw}{\cctr@tempxw}
745 \DIVIDE{#1}{\cctr@tempxw}{\cctr@tempxw}
746 \ADD{\cctr@tempw}{\cctr@tempxw}{\cctr@tempw}
747 \SUBTRACT{\cctr@tempw}{1}{\cctr@tempw}
748 \SUBTRACT{\cctr@tempw}{\cctr@tempoldw}{\cctr@tempdif}
749 \cctr@lengthb=\cctr@tempdif\p@%
750 \ifnum
751 \cctr@lengthb<\z@ \cctr@lengthb=-\cctr@lengthb
752 \fi}%
753 \COPY{\cctr@tempw}{#2}\@OUTPUTSOL{#2}}

Inverse trigonometric functions

\ARCSIN \ARCSIN{h#1 i}{h#2 i} defines #2 as the arcsin of #1, using the Newton’s method: xn+1 =
xn − (sin xn − #1)/(cos xn ).
754 \def\ARCSIN#1#2{%
755 \begingroup
756 \ifdim #1\p@ = \z@
757 \COPY{0}{#2}
758 \else
759 \ifdim #1\p@ = 1\p@
760 \COPY{\numberHALFPI}{#2}
761 \else
762 \ifdim #1\p@ = -1\p@

49
763 \COPY{-\numberHALFPI}{#2}
764 \else
765 \ifdim #1\p@ > 1\p@
766 \let#2\undefined
767 \cctr@Warnbigarcsin{#1}
768 \else
769 \ifdim #1\p@ < -1\p@
770 \let#2\undefined
771 \cctr@Warnbigarcsin{#1}
772 \else
p
If x is close to 1 we use arcsin x = π/2 − 2 arcsin (1 − x)/2
773 \ifdim #1\p@ >0.89\p@
774 \SUBTRACT{1}{#1}{\cctr@tempx}
775 \DIVIDE{\cctr@tempx}{2}{\cctr@tempx}
776 \SQRT{\cctr@tempx}{\cctr@tempxx}
777 \ARCSIN{\cctr@tempxx}{#2}
778 \MULTIPLY{2}{#2}{#2}
779 \SUBTRACT{\numberHALFPI}{#2}{#2}
780 \else
p
Symmetrically, for x close to −1, arcsin x = −π/2 + 2 arcsin (1 + x)/2
781 \ifdim #1\p@ <-0.89\p@
782 \ADD{1}{#1}{\cctr@tempx}
783 \DIVIDE{\cctr@tempx}{2}{\cctr@tempx}
784 \SQRT{\cctr@tempx}{\cctr@tempxx}
785 \ARCSIN{\cctr@tempxx}{#2}
786 \MULTIPLY{2}{#2}{#2}
787 \SUBTRACT{#2}{\numberHALFPI}{#2}
788 \else
We take #1 as the initial approximation.
789 \COPY{#1}{#2}
If −0.4 ≤ t ≤ 0.4 then arcsin x ≈ x is a good approximation. Else, we apply the Newton method
790 \ABSVALUE{#1}{\cctr@tempy}
791 \ifdim \cctr@tempy\p@ < 0.04\p@
792 \else
\cctr@lengthb will be the difference of two successive iterations, and \cctr@tempoldy,
\cctr@tempy will be the two last iterations.
We start with \cctr@lengthb=5\p@ and \cctr@tempy=16383 to ensure almost one iteration.
793 \cctr@lengthb=5\p@
794 \COPY{16383}{\cctr@tempy}
Successive iterations
795 \@whilenum \cctr@lengthb>\cctr@epsilon \do {%
Copy the actual approximation to \cctr@tempw
796 \COPY{#2}{\cctr@tempw}
797 \COPY{\cctr@tempy}{\cctr@tempoldy}
798 \SIN{\cctr@tempw}{\cctr@tempz}
799 \SUBTRACT{\cctr@tempz}{#1}{\cctr@tempz}

50
 800 \COS{\cctr@tempw}{\cctr@tempy}
801 \DIVIDE{\cctr@tempz}{\cctr@tempy}{\cctr@tempz}
802 \SUBTRACT{\cctr@tempw}{\cctr@tempz}{\cctr@tempz}
Now, \cctr@tempz is the new approximation.
803 \COPY{\cctr@tempz}{#2}
Finally, we store in \cctr@lengthb the difference of the two last approximations, finishing the
loop.
804 \SUBTRACT{#2}{\cctr@tempw}{\cctr@tempy}
805 \ABSVALUE{\cctr@tempy}{\cctr@tempy}
806 \cctr@lengthb=\cctr@tempy\p@%
807 \ifdim\cctr@tempy\p@=\cctr@tempoldy\p@
808 \cctr@lengthb=\z@
809 \fi}\fi\fi\fi\fi\fi\fi\fi\fi\@OUTPUTSOL{#2}}

\ARCCOS \ARCCOS{h#1 i}{h#2 i} defines #2 as the arccos of #1, using the well know relation arccos x =
π/2 − arcsin x.
810 \def\ARCCOS#1#2{%
811 \begingroup
812 \ifdim #1\p@ = \z@
813 \COPY{\numberHALFPI}{#2}
814 \else
815 \ifdim #1\p@ = 1\p@
816 \COPY{0}{#2}
817 \else
818 \ifdim #1\p@ = -1\p@
819 \COPY{\numberPI}{#2}
820 \else
821 \ifdim #1\p@ > 1\p@
822 \let#2\undefined
823 \cctr@Warnbigarccos{#1}
824 \else
825 \ifdim #1\p@ < -1\p@
826 \let#2\undefined
827 \cctr@Warnbigarccos{#1}
828 \else
829 \ARCSIN{#1}{#2}
830 \SUBTRACT{\numberHALFPI}{#2}{#2}
831 \fi\fi\fi\fi\fi\@OUTPUTSOL{#2}}

\ARCTAN \ARCTAN{h#1 i}{h#2 i}. arctan of #1.

832 \def\ARCTAN#1#2{%
833 \begingroup
If |t| > 1, compute arctan x using arctan x = sign(x)π/2 − arctan(1/x).
834 \ifdim#1\p@<-1\p@
835 \DIVIDE{1}{#1}{\cctr@tempb}
836 \ARCTAN{\cctr@tempb}{#2}
837 \SUBTRACT{-\numberHALFPI}{#2}{#2}
838 \else

51
 839 \ifdim#1\p@>1\p@
840 \DIVIDE{1}{#1}{\cctr@tempb}
841 \ARCTAN{\cctr@tempb}{#2}
842 \SUBTRACT{\numberHALFPI}{#2}{#2}
843 \else
For −1 ≤ x ≤ 1 call \@BASICARCTAN.
844 \@BASICARCTAN{#1}{#2}
845 \fi
846 \fi\@OUTPUTSOL{#2}}

\@BASICARCTAN \@BASICARCTAN{h#1 i}{h#2 i} applies this approximation:

x
arctan x =
x2
1+
(2x)2
3+
(3x)2
5+
(4x)2
7+
9 + ···
847 \def\@BASICARCTAN#1#2{%
848 \begingroup
Exact arctan of zero
849 \ifdim#1\p@=\z@ \COPY{0}{#2}
850 \else
Compute the continued fraction.
851 \SQUARE{#1}{\cctr@tempa}
852 \MULTIPLY{64}{\cctr@tempa}{#2}
853 \ADD{15}{#2}{#2}
854 \DIVIDE{\cctr@tempa}{#2}{#2}
855 \MULTIPLY{49}{#2}{#2}
856 \ADD{13}{#2}{#2}
857 \DIVIDE{\cctr@tempa}{#2}{#2}
858 \MULTIPLY{36}{#2}{#2}
859 \ADD{11}{#2}{#2}
860 \DIVIDE{\cctr@tempa}{#2}{#2}
861 \MULTIPLY{25}{#2}{#2}
862 \ADD{9}{#2}{#2}
863 \DIVIDE{\cctr@tempa}{#2}{#2}
864 \MULTIPLY{16}{#2}{#2}
865 \ADD{7}{#2}{#2}
866 \DIVIDE{\cctr@tempa}{#2}{#2}
867 \MULTIPLY{9}{#2}{#2}
868 \ADD{5}{#2}{#2}
869 \DIVIDE{\cctr@tempa}{#2}{#2}
870 \MULTIPLY{4}{#2}{#2}
871 \ADD{3}{#2}{#2}
872 \DIVIDE{\cctr@tempa}{#2}{#2}
873 \ADD{1}{#2}{#2}

52
 874 \DIVIDE{#1}{#2}{#2}
875 \fi\@OUTPUTSOL{#2}}

\ARCCOT \ARCCOT{h#1 i}{h#2 i} defines #2 as the arccot of #1, using the well know relation arccot x =
π/2 − arctan x.
876 \def\ARCCOT#1#2{%
877 \begingroup
878 \ARCTAN{#1}{#2}
879 \SUBTRACT{\numberHALFPI}{#2}{#2}
880 \@OUTPUTSOL{#2}}

Inverse hyperbolic functions

√

\ARSINH \ARSINH{h#1 i}{h#2 i}. Inverse hyperbolic sine of #1 : arsinh x = log x + 1 + x2
881 \def\ARSINH#1#2{%
882 \begingroup
883 \SQUARE{#1}{\cctr@tempa}
884 \ADD{1}{\cctr@tempa}{\cctr@tempa}
885 \SQRT{\cctr@tempa}{\cctr@tempb}
886 \ADD{#1}{\cctr@tempb}{\cctr@tempb}
887 \LOG\cctr@tempb{#2}
888 \@OUTPUTSOL{#2}}
√

\ARCOSH \ARCOSH{h#1 i}{h#2 i}. Inverse hyperbolic sine of #1 : arcosh x = log x + x2 − 1
889 \def\ARCOSH#1#2{%
890 \begingroup
If x < 1, this function is no defined
891 \ifdim#1\p@<1\p@
892 \let#2\undefined
893 \cctr@Warnsmallarcosh{#1}
894 \else
895 \SQUARE{#1}{\cctr@tempa}
896 \SUBTRACT{\cctr@tempa}{1}{\cctr@tempa}
897 \SQRT{\cctr@tempa}{\cctr@tempb}
898 \ADD{#1}{\cctr@tempb}{\cctr@tempb}
899 \LOG\cctr@tempb{#2}
900 \fi\@OUTPUTSOL{#2}}
1
\ARTANH \ARTANH{h#1 i}{h#2 i}. Inverse hyperbolic tangent of #1 : artanh x = 2 log ((1 + x) − log(1 − x))
901 \def\ARTANH#1#2{%
902 \begingroup
If |x| ≥ 1, this function is no defined
903 \ifdim#1\p@<-0.99998\p@
904 \let#2\undefined
905 \cctr@Warnbigartanh{#1}
906 \else
907 \ifdim#1\p@>0.99998\p@
908 \let#2\undefined

53
 909 \cctr@Warnbigartanh{#1}
910 \else
911 \COPY{#1}{\cctr@tempa}
912 \ADD1\cctr@tempa\cctr@tempb
913 \SUBTRACT1\cctr@tempa\cctr@tempc
914 \LOG\cctr@tempb\cctr@tempB
915 \LOG\cctr@tempc\cctr@tempC
916 \SUBTRACT\cctr@tempB\cctr@tempC{#2}
917 \DIVIDE{#2}{2}{#2}
918 \fi
919 \fi\@OUTPUTSOL{#2}}

\ARCOTH \ARCOTH{h#1 i}{h#2 i}. Inverse hyperbolic cotangent of #1 :

arcoth x = sign(x) 12 log ((x + 1) − log(x − 1))
920 \def\ARCOTH#1#2{%
921 \begingroup
If |x| ≤ 1, this function is no defined
922 \ifdim#1\p@>-0.99998\p@
923 \ifdim#1\p@<0.99998\p@
924 \let#2\undefined
925 \cctr@Warnsmallarcoth{#1}
926 \else
927 \ifdim#1\p@>\p@
1
For x > 1, calcule arcoth x = 2 log ((x + 1) − log(x − 1))
928 \COPY{#1}{\cctr@tempa}
929 \ADD1\cctr@tempa\cctr@tempb
930 \SUBTRACT\cctr@tempa1\cctr@tempc
931 \LOG\cctr@tempb\cctr@tempB
932 \LOG\cctr@tempc\cctr@tempC
933 \SUBTRACT\cctr@tempB\cctr@tempC{#2}
934 \DIVIDE{#2}{2}{#2}
935 \else
936 \fi
937 \fi
938 \else
For x < −1, calcule − artanh(−x)
939 \MULTIPLY{-1}{#1}{\cctr@tempa}
940 \ARCOTH{\cctr@tempa}{#2}
941 \COPY{-#2}{#2}
942 \fi\@OUTPUTSOL{#2}}

13.4 Matrix arithmetics

Vector operations
\VECTORSIZE The size of a vector is 2 or 3. \VECTORSIZE(h#1 i){h#2 i} stores in #2 the size of (h#1 i).
Almost all vector commands needs to know the vector size.
943 \def\VECTORSIZE(#1)#2{\expandafter\@VECTORSIZE(#1,,){#2}}

54
 944 \def\@VECTORSIZE(#1,#2,#3,#4)#5{\ifx$#3$\COPY{2}{#5}
945 \else\COPY{3}{#5}\fi\ignorespaces}

\VECTORCOPY \VECTORCOPY(h#1,#2 i)(h#3,#4 i) stores #1 and #2 in #3 and #4.

\VECTORCOPY(h#1,#2,#3 i)(h#4,#5#6 i) stores #1, #2 and #3 in #4 and #5 and #6.
946 \def\@@VECTORCOPY(#1,#2)(#3,#4){%
947 \COPY{#1}{#3}\COPY{#2}{#4}}
948
949 \def\@@@VECTORCOPY(#1,#2,#3)(#4,#5,#6){%
950 \COPY{#1}{#4}\COPY{#2}{#5}\COPY{#3}{#6}}
951
952 \def\VECTORCOPY(#1)(#2){%
953 \VECTORSIZE(#1){\cctr@size}
954 \ifnum\cctr@size=2
955 \@@VECTORCOPY(#1)(#2)
956 \else \@@@VECTORCOPY(#1)(#2)\fi}

\VECTORGLOBALCOPY \VECTORGLOBALCOPY is the global version of \VECTORCOPY

957 \def\@@VECTORGLOBALCOPY(#1,#2)(#3,#4){%
958 \GLOBALCOPY{#1}{#3}\GLOBALCOPY{#2}{#4}}
959
960 \def\@@@VECTORGLOBALCOPY(#1,#2,#3)(#4,#5,#6){%
961 \GLOBALCOPY{#1}{#4}\GLOBALCOPY{#2}{#5}\GLOBALCOPY{#3}{#6}}
962
963 \def\VECTORGLOBALCOPY(#1)(#2){%
964 \VECTORSIZE(#1){\cctr@size}
965 \ifnum\cctr@size=2
966 \@@VECTORGLOBALCOPY(#1)(#2)
967 \else \@@@VECTORGLOBALCOPY(#1)(#2)\fi}

\@OUTPUTVECTOR
968 \def\@@OUTPUTVECTOR(#1,#2){%
969 \VECTORGLOBALCOPY(#1,#2)(\cctr@outa,\cctr@outb)
970 \endgroup\VECTORCOPY(\cctr@outa,\cctr@outb)(#1,#2)}
971
972 \def\@@@OUTPUTVECTOR(#1,#2,#3){%
973 \VECTORGLOBALCOPY(#1,#2,#3)(\cctr@outa,\cctr@outb,\cctr@outc)
974 \endgroup\VECTORCOPY(\cctr@outa,\cctr@outb,\cctr@outc)(#1,#2,#3)}
975
976 \def\@OUTPUTVECTOR(#1){\VECTORSIZE(#1){\cctr@size}
977 \ifnum\cctr@size=2
978 \@@OUTPUTVECTOR(#1)
979 \else \@@@OUTPUTVECTOR(#1)\fi}

\SCALARPRODUCT Scalar product of two vectors.

980 \def\@@SCALARPRODUCT(#1,#2)(#3,#4)#5{%
981 \MULTIPLY{#1}{#3}{#5}
982 \MULTIPLY{#2}{#4}\cctr@tempa
983 \ADD{#5}{\cctr@tempa}{#5}}
984

55
 985 \def\@@@SCALARPRODUCT(#1,#2,#3)(#4,#5,#6)#7{%
986 \MULTIPLY{#1}{#4}{#7}
987 \MULTIPLY{#2}{#5}\cctr@tempa
988 \ADD{#7}{\cctr@tempa}{#7}
989 \MULTIPLY{#3}{#6}\cctr@tempa
990 \ADD{#7}{\cctr@tempa}{#7}}
991
992 \def\SCALARPRODUCT(#1)(#2)#3{%
993 \begingroup
994 \VECTORSIZE(#1){\cctr@size}
995 \ifnum\cctr@size=2
996 \@@SCALARPRODUCT(#1)(#2){#3}
997 \else \@@@SCALARPRODUCT(#1)(#2){#3}\fi\@OUTPUTSOL{#3}}

\DOTPRODUCT \DOTPRODUCT is an alias for \SCALARPRODUCT.

998 \let\DOTPRODUCT\SCALARPRODUCT

\VECTORPRODUCT Vector product of two (three dimensional) vectors.

999 \def\@@VECTORPRODUCT(#1)(#2)(#3,#4){%
1000 \let#3\undefined
1001 \let#4\undefined
1002 \cctr@Warncrossprod(#1)(#2)}
1003
1004 \def\@@@VECTORPRODUCT(#1,#2,#3)(#4,#5,#6)(#7,#8,#9){%
1005 \DETERMINANT(#2,#3;#5,#6){#7}
1006 \DETERMINANT(#3,#1;#6,#4){#8}
1007 \DETERMINANT(#1,#2;#4,#5){#9}}
1008
1009 \def\VECTORPRODUCT(#1)(#2)(#3){%
1010 \begingroup
1011 \VECTORSIZE(#1){\cctr@size}
1012 \ifnum\cctr@size=2
1013 \@@VECTORPRODUCT(#1)(#2)(#3)
1014 \else \@@@VECTORPRODUCT(#1)(#2)(#3)\fi\@OUTPUTSOL{#3}}

\CROSSPRODUCT \CROSSPRODUCT is an alias for \VECTORPRODUCT.

1015 \let\CROSSPRODUCT\VECTORPRODUCT

\VECTORADD Sum of two vectors.

1016 \def\@@VECTORADD(#1,#2)(#3,#4)(#5,#6){%
1017 \ADD{#1}{#3}{#5}
1018 \ADD{#2}{#4}{#6}}
1019
1020 \def\@@@VECTORADD(#1,#2,#3)(#4,#5,#6)(#7,#8,#9){%
1021 \ADD{#1}{#4}{#7}
1022 \ADD{#2}{#5}{#8}
1023 \ADD{#3}{#6}{#9}}
1024
1025 \def\VECTORADD(#1)(#2)(#3){%

56
 1026 \VECTORSIZE(#1){\cctr@size}
1027 \ifnum\cctr@size=2
1028 \@@VECTORADD(#1)(#2)(#3)
1029 \else \@@@VECTORADD(#1)(#2)(#3)\fi}

\VECTORSUB Difference of two vectors.

1030 \def\@@VECTORSUB(#1,#2)(#3,#4)(#5,#6){%
1031 \VECTORADD(#1,#2)(-#3,-#4)(#5,#6)}
1032
1033 \def\@@@VECTORSUB(#1,#2,#3)(#4,#5,#6)(#7,#8,#9){%
1034 \VECTORADD(#1,#2,#3)(-#4,-#5,-#6)(#7,#8,#9)}
1035
1036 \def\VECTORSUB(#1)(#2)(#3){%
1037 \VECTORSIZE(#1){\cctr@size}
1038 \ifnum\cctr@size=2
1039 \@@VECTORSUB(#1)(#2)(#3)
1040 \else \@@@VECTORSUB(#1)(#2)(#3)\fi}

\VECTORABSVALUE Absolute value of a each entry of a vector.

1041 \def\@@VECTORABSVALUE(#1,#2)(#3,#4){%
1042 \ABSVALUE{#1}{#3}\ABSVALUE{#2}{#4}}
1043
1044 \def\@@@VECTORABSVALUE(#1,#2,#3)(#4,#5,#6){%
1045 \ABSVALUE{#1}{#4}\ABSVALUE{#2}{#5}\ABSVALUE{#3}{#6}}
1046
1047 \def\VECTORABSVALUE(#1)(#2){%
1048 \VECTORSIZE(#1){\cctr@size}
1049 \ifnum\cctr@size=2
1050 \@@VECTORABSVALUE(#1)(#2)
1051 \else \@@@VECTORABSVALUE(#1)(#2)\fi}

\SCALARVECTORPRODUCT Scalar-vector product.

1052 \def\@@SCALARVECTORPRODUCT#1(#2,#3)(#4,#5){%
1053 \MULTIPLY{#1}{#2}{#4}
1054 \MULTIPLY{#1}{#3}{#5}}
1055
1056 \def\@@@SCALARVECTORPRODUCT#1(#2,#3,#4)(#5,#6,#7){%
1057 \MULTIPLY{#1}{#2}{#5}
1058 \MULTIPLY{#1}{#3}{#6}
1059 \MULTIPLY{#1}{#4}{#7}}
1060
1061 \def\SCALARVECTORPRODUCT#1(#2)(#3){%
1062 \VECTORSIZE(#2){\cctr@size}
1063 \ifnum\cctr@size=2
1064 \@@SCALARVECTORPRODUCT{#1}(#2)(#3)
1065 \else \@@@SCALARVECTORPRODUCT{#1}(#2)(#3)\fi}

\VECTORNORM Euclidean norm of a vector.

1066 \def\VECTORNORM(#1)#2{%
1067 \begingroup

57
 1068 \SCALARPRODUCT(#1)(#1){\cctr@temp}
1069 \SQUAREROOT{\cctr@temp}{#2}\@OUTPUTSOL{#2}}

\UNITVECTOR Unitary vector parallel to a given vector.

1070 \def\UNITVECTOR(#1)(#2){%
1071 \begingroup
1072 \VECTORNORM(#1){\cctr@tempa}
1073 \DIVIDE{1}{\cctr@tempa}{\cctr@tempa}
1074 \SCALARVECTORPRODUCT{\cctr@tempa}(#1)(#2)\@OUTPUTVECTOR(#2)}

\TWOVECTORSANGLE Angle between two vectors.

1075 \def\TWOVECTORSANGLE(#1)(#2)#3{%
1076 \begingroup
1077 \VECTORNORM(#1){\cctr@tempa}
1078 \VECTORNORM(#2){\cctr@tempb}
1079 \SCALARPRODUCT(#1)(#2){\cctr@tempc}
1080 \ifdim \cctr@tempa\p@ =\z@
1081 \let#3\undefined
1082 \cctr@Warnnoangle(#1)(#2)
1083 \else
1084 \ifdim \cctr@tempb\p@ =\z@
1085 \let#3\undefined
1086 \cctr@Warnnoangle(#1)(#2)
1087 \else
1088 \DIVIDE{\cctr@tempc}{\cctr@tempa}{\cctr@tempc}
1089 \DIVIDE{\cctr@tempc}{\cctr@tempb}{\cctr@tempc}
1090 \ARCCOS{\cctr@tempc}{#3}
1091 \fi\fi\@OUTPUTSOL{#3}}

Matrix operations
Here, we need to define some internal macros to simulate commands with more than nine
arguments.
\@TDMATRIXCOPY This command copies a 3 × 3 matrix to the commands \cctr@solAA, \cctr@solAB, . . . ,
\cctr@solCC.
1092 \def\@TDMATRIXCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1093 \COPY{#1}{\cctr@solAA}
1094 \COPY{#2}{\cctr@solAB}
1095 \COPY{#3}{\cctr@solAC}
1096 \COPY{#4}{\cctr@solBA}
1097 \COPY{#5}{\cctr@solBB}
1098 \COPY{#6}{\cctr@solBC}
1099 \COPY{#7}{\cctr@solCA}
1100 \COPY{#8}{\cctr@solCB}
1101 \COPY{#9}{\cctr@solCC}}

\@TDMATRIXSOL This command copies the commands \cctr@solAA, \cctr@solAB, . . . , \cctr@solCC to a 3 × 3

matrix. This macro is used to store the results of a matrix operation.
1102 \def\@TDMATRIXSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9){%

58
 1103 \COPY{\cctr@solAA}{#1}
1104 \COPY{\cctr@solAB}{#2}
1105 \COPY{\cctr@solAC}{#3}
1106 \COPY{\cctr@solBA}{#4}
1107 \COPY{\cctr@solBB}{#5}
1108 \COPY{\cctr@solBC}{#6}
1109 \COPY{\cctr@solCA}{#7}
1110 \COPY{\cctr@solCB}{#8}
1111 \COPY{\cctr@solCC}{#9}}

\@TDMATRIXGLOBALSOL
1112 \def\@TDMATRIXGLOBALSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1113 \GLOBALCOPY{\cctr@solAA}{#1}
1114 \GLOBALCOPY{\cctr@solAB}{#2}
1115 \GLOBALCOPY{\cctr@solAC}{#3}
1116 \GLOBALCOPY{\cctr@solBA}{#4}
1117 \GLOBALCOPY{\cctr@solBB}{#5}
1118 \GLOBALCOPY{\cctr@solBC}{#6}
1119 \GLOBALCOPY{\cctr@solCA}{#7}
1120 \GLOBALCOPY{\cctr@solCB}{#8}
1121 \GLOBALCOPY{\cctr@solCC}{#9}}

\@TDMATRIXNOSOL This command undefines a 3 × 3 matrix when a matrix problem has no solution.
1122 \def\@TDMATRIXNOSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1123 \let#1\undefined
1124 \let#2\undefined
1125 \let#3\undefined
1126 \let#4\undefined
1127 \let#5\undefined
1128 \let#6\undefined
1129 \let#7\undefined
1130 \let#8\undefined
1131 \let#9\undefined
1132 }

\@@TDMATRIXSOL This command stores or undefines the solution.

1133 \def\@@TDMATRIXSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1134 \ifx\cctr@solAA\undefined
1135 \@TDMATRIXNOSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9)%
1136 \else
1137 \@TDMATRIXSOL(#1,#2,#3;#4,#5,#6;#7,#8,#9)\fi}

\@NUMBERSOL This command stores the scalar solution of a matrix operation.

1138 \def\@NUMBERSOL#1{\COPY{\cctr@sol}{#1}}

\MATRIXSIZE Size (2 or 3) of a matrix.

1139 \def\MATRIXSIZE(#1)#2{\expandafter\@MATRIXSIZE(#1;;){#2}}
1140 \def\@MATRIXSIZE(#1;#2;#3;#4)#5{\ifx$#3$\COPY{2}{#5}
1141 \else\COPY{3}{#5}\fi\ignorespaces}

59
 \MATRIXCOPY Store a matrix in 4 or 9 commands.
1142 \def\@@MATRIXCOPY(#1,#2;#3,#4)(#5,#6;#7,#8){%
1143 \COPY{#1}{#5}\COPY{#2}{#6}\COPY{#3}{#7}\COPY{#4}{#8}}
1144
1145 \def\@@@MATRIXCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1146 \@TDMATRIXCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9)
1147 \@TDMATRIXSOL}
1148
1149 \def\MATRIXCOPY(#1)(#2){%
1150 \MATRIXSIZE(#1){\cctr@size}
1151 \ifnum\cctr@size=2
1152 \@@MATRIXCOPY(#1)(#2)
1153 \else \@@@MATRIXCOPY(#1)(#2)\fi}

\MATRIXGLOBALCOPY Global version of \MATRIXCOPY.

1154 \def\@@MATRIXGLOBALCOPY(#1,#2;#3,#4)(#5,#6;#7,#8){%
1155 \GLOBALCOPY{#1}{#5}\GLOBALCOPY{#2}{#6}\GLOBALCOPY{#3}{#7}\GLOBALCOPY{#4}{#8}}
1156
1157 \def\@@@MATRIXGLOBALCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1158 \@TDMATRIXCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9)
1159 \@TDMATRIXGLOBALSOL}
1160
1161 \def\MATRIXGLOBALCOPY(#1)(#2){%
1162 \MATRIXSIZE(#1){\cctr@size}
1163 \ifnum\cctr@size=2
1164 \@@MATRIXGLOBALCOPY(#1)(#2)
1165 \else \@@@MATRIXGLOBALCOPY(#1)(#2)\fi}

\@OUTPUTMATRIX
1166 \def\@@OUTPUTMATRIX(#1,#2;#3,#4){%
1167 \MATRIXGLOBALCOPY(#1,#2;#3,#4)(\cctr@outa,\cctr@outb;\cctr@outc,\cctr@outd)
1168 \endgroup\MATRIXCOPY(\cctr@outa,\cctr@outb;\cctr@outc,\cctr@outd)(#1,#2;#3,#4)}
1169
1170 \def\@@@OUTPUTMATRIX(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1171 \MATRIXGLOBALCOPY(#1,#2,#3;#4,#5,#6;#7,#8,#9)(%
1172 \cctr@outa,\cctr@outb,\cctr@outc;
1173 \cctr@outd,\cctr@oute,\cctr@outf;
1174 \cctr@outg,\cctr@outh,\cctr@outi)
1175 \endgroup\MATRIXCOPY(%
1176 \cctr@outa,\cctr@outb,\cctr@outc;
1177 \cctr@outd,\cctr@oute,\cctr@outf;
1178 \cctr@outg,\cctr@outh,\cctr@outi)(#1,#2,#3;#4,#5,#6;#7,#8,#9)}
1179
1180 \def\@OUTPUTMATRIX(#1){\MATRIXSIZE(#1){\cctr@size}
1181 \ifnum\cctr@size=2
1182 \@@OUTPUTMATRIX(#1)
1183 \else \@@@OUTPUTMATRIX(#1)\fi}

\TRANSPOSEMATRIX Matrix transposition.

1184 \def\@@TRANSPOSEMATRIX(#1,#2;#3,#4)(#5,#6;#7,#8){%

60
 1185 \COPY{#1}{#5}\COPY{#3}{#6}\COPY{#2}{#7}\COPY{#4}{#8}}
1186
1187 \def\@@@TRANSPOSEMATRIX(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1188 \@TDMATRIXCOPY(#1,#4,#7;#2,#5,#8;#3,#6,#9)
1189 \@TDMATRIXSOL}
1190
1191 \def\TRANSPOSEMATRIX(#1)(#2){%
1192 \begingroup
1193 \MATRIXSIZE(#1){\cctr@size}
1194 \ifnum\cctr@size=2
1195 \@@TRANSPOSEMATRIX(#1)(#2)
1196 \else \@@@TRANSPOSEMATRIX(#1)(#2)\fi\@OUTPUTMATRIX(#2)}

\MATRIXADD Sum of two matrices.

1197 \def\@@MATRIXADD(#1;#2)(#3;#4)(#5,#6;#7,#8){%
1198 \VECTORADD(#1)(#3)(#5,#6)
1199 \VECTORADD(#2)(#4)(#7,#8)}
1200
1201 \def\@@@MATRIXADD(#1;#2;#3)(#4;#5;#6){%
1202 \VECTORADD(#1)(#4)(\cctr@solAA,\cctr@solAB,\cctr@solAC)
1203 \VECTORADD(#2)(#5)(\cctr@solBA,\cctr@solBB,\cctr@solBC)
1204 \VECTORADD(#3)(#6)(\cctr@solCA,\cctr@solCB,\cctr@solCC)
1205 \@TDMATRIXSOL}
1206
1207 \def\MATRIXADD(#1)(#2)(#3){%
1208 \begingroup
1209 \MATRIXSIZE(#1){\cctr@size}
1210 \ifnum\cctr@size=2
1211 \@@MATRIXADD(#1)(#2)(#3)
1212 \else \@@@MATRIXADD(#1)(#2)(#3)\fi\@OUTPUTMATRIX(#3)}

\MATRIXSUB Difference of two matrices.

1213 \def\@@MATRIXSUB(#1;#2)(#3;#4)(#5,#6;#7,#8){%
1214 \VECTORSUB(#1)(#3)(#5,#6)
1215 \VECTORSUB(#2)(#4)(#7,#8)}
1216
1217 \def\@@@MATRIXSUB(#1;#2;#3)(#4;#5;#6){%
1218 \VECTORSUB(#1)(#4)(\cctr@solAA,\cctr@solAB,\cctr@solAC)
1219 \VECTORSUB(#2)(#5)(\cctr@solBA,\cctr@solBB,\cctr@solBC)
1220 \VECTORSUB(#3)(#6)(\cctr@solCA,\cctr@solCB,\cctr@solCC)
1221 \@TDMATRIXSOL}
1222
1223 \def\MATRIXSUB(#1)(#2)(#3){%
1224 \begingroup
1225 \MATRIXSIZE(#1){\cctr@size}
1226 \ifnum\cctr@size=2
1227 \@@MATRIXSUB(#1)(#2)(#3)
1228 \else \@@@MATRIXSUB(#1)(#2)(#3)\fi\@OUTPUTMATRIX(#3)}

\MATRIXABSVALUE Absolute value (of each entry) of a matrix.

61
 1229 \def\@@MATRIXABSVALUE(#1;#2)(#3;#4){%
1230 \VECTORABSVALUE(#1)(#3)\VECTORABSVALUE(#2)(#4)}
1231
1232 \def\@@@MATRIXABSVALUE(#1;#2;#3)(#4;#5;#6){%
1233 \VECTORABSVALUE(#1)(#4)\VECTORABSVALUE(#2)(#5)\VECTORABSVALUE(#3)(#6)}
1234
1235 \def\MATRIXABSVALUE(#1)(#2){%
1236 \begingroup
1237 \MATRIXSIZE(#1){\cctr@size}
1238 \ifnum\cctr@size=2
1239 \@@MATRIXABSVALUE(#1)(#2)
1240 \else \@@@MATRIXABSVALUE(#1)(#2)\fi\@OUTPUTMATRIX(#2)}

\MATRIXVECTORPRODUCT Matrix-vector product.

1241 \def\@@MATRIXVECTORPRODUCT(#1;#2)(#3)(#4,#5){%
1242 \SCALARPRODUCT(#1)(#3){#4}
1243 \SCALARPRODUCT(#2)(#3){#5}}
1244
1245 \def\@@@MATRIXVECTORPRODUCT(#1;#2;#3)(#4)(#5,#6,#7){%
1246 \SCALARPRODUCT(#1)(#4){#5}
1247 \SCALARPRODUCT(#2)(#4){#6}
1248 \SCALARPRODUCT(#3)(#4){#7}}
1249
1250 \def\MATRIXVECTORPRODUCT(#1)(#2)(#3){%
1251 \begingroup
1252 \MATRIXSIZE(#1){\cctr@size}
1253 \ifnum\cctr@size=2
1254 \@@MATRIXVECTORPRODUCT(#1)(#2)(#3)
1255 \else \@@@MATRIXVECTORPRODUCT(#1)(#2)(#3)\fi\@OUTPUTVECTOR(#3)}

\VECTORMATRIXPRODUCT Vector-matrix product.

1256 \def\@@VECTORMATRIXPRODUCT(#1)(#2,#3;#4,#5)(#6,#7){%
1257 \SCALARPRODUCT(#1)(#2,#4){#6}
1258 \SCALARPRODUCT(#1)(#3,#5){#7}}
1259
1260 \def\@@@VECTORMATRIXPRODUCT(#1,#2,#3)(#4;#5;#6)(#7){%
1261 \SCALARVECTORPRODUCT{#1}(#4)(#7)
1262 \SCALARVECTORPRODUCT{#2}(#5)(\cctr@tempa,\cctr@tempb,\cctr@tempc)
1263 \VECTORADD(#7)(\cctr@tempa,\cctr@tempb,\cctr@tempc)(#7)
1264 \SCALARVECTORPRODUCT{#3}(#6)(\cctr@tempa,\cctr@tempb,\cctr@tempc)
1265 \VECTORADD(#7)(\cctr@tempa,\cctr@tempb,\cctr@tempc)(#7)}
1266
1267 \def\VECTORMATRIXPRODUCT(#1)(#2)(#3){%
1268 \begingroup
1269 \VECTORSIZE(#1){\cctr@size}
1270 \ifnum\cctr@size=2
1271 \@@VECTORMATRIXPRODUCT(#1)(#2)(#3)
1272 \else \@@@VECTORMATRIXPRODUCT(#1)(#2)(#3)\fi\@OUTPUTVECTOR(#3)}

\SCALARMATRIXPRODUCT Scalar-matrix product.

62
 1273 \def\@@SCALARMATRIXPRODUCT#1(#2;#3)(#4,#5;#6,#7){%
1274 \SCALARVECTORPRODUCT{#1}(#2)(#4,#5)
1275 \SCALARVECTORPRODUCT{#1}(#3)(#6,#7)}
1276
1277 \def\@@@SCALARMATRIXPRODUCT#1(#2;#3;#4){%
1278 \SCALARVECTORPRODUCT{#1}(#2)(\cctr@solAA,\cctr@solAB,\cctr@solAC)
1279 \SCALARVECTORPRODUCT{#1}(#3)(\cctr@solBA,\cctr@solBB,\cctr@solBC)
1280 \SCALARVECTORPRODUCT{#1}(#4)(\cctr@solCA,\cctr@solCB,\cctr@solCC)
1281 \@TDMATRIXSOL}
1282
1283 \def\SCALARMATRIXPRODUCT#1(#2)(#3){%
1284 \begingroup
1285 \MATRIXSIZE(#2){\cctr@size}
1286 \ifnum\cctr@size=2
1287 \@@SCALARMATRIXPRODUCT{#1}(#2)(#3)
1288 \else \@@@SCALARMATRIXPRODUCT{#1}(#2)(#3)\fi\@OUTPUTMATRIX(#3)}

\MATRIXPRODUCT Product of two matrices.

1289 \def\@@MATRIXPRODUCT(#1)(#2,#3;#4,#5)(#6,#7;#8,#9){%
1290 \MATRIXVECTORPRODUCT(#1)(#2,#4)(#6,#8)
1291 \MATRIXVECTORPRODUCT(#1)(#3,#5)(#7,#9)}
1292
1293 \def\@@@MATRIXPRODUCT(#1;#2;#3)(#4){%
1294 \VECTORMATRIXPRODUCT(#1)(#4)(\cctr@solAA,\cctr@solAB,\cctr@solAC)
1295 \VECTORMATRIXPRODUCT(#2)(#4)(\cctr@solBA,\cctr@solBB,\cctr@solBC)
1296 \VECTORMATRIXPRODUCT(#3)(#4)(\cctr@solCA,\cctr@solCB,\cctr@solCC)
1297 \@TDMATRIXSOL}
1298
1299 \def\MATRIXPRODUCT(#1)(#2)(#3){%
1300 \begingroup
1301 \MATRIXSIZE(#1){\cctr@size}
1302 \ifnum\cctr@size=2
1303 \@@MATRIXPRODUCT(#1)(#2)(#3)
1304 \else \@@@MATRIXPRODUCT(#1)(#2)(#3)\fi\@OUTPUTMATRIX(#3)}

\DETERMINANT Determinant of a matrix.

1305 \def\@@DETERMINANT(#1,#2;#3,#4)#5{%
1306 \MULTIPLY{#1}{#4}{#5}
1307 \MULTIPLY{#2}{#3}{\cctr@tempa}
1308 \SUBTRACT{#5}{\cctr@tempa}{#5}}
1309
1310 \def\@@@DETERMINANT(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1311 \DETERMINANT(#5,#6;#8,#9){\cctr@det}\MULTIPLY{#1}{\cctr@det}{\cctr@sol}
1312 \DETERMINANT(#6,#4;#9,#7){\cctr@det}\MULTIPLY{#2}{\cctr@det}{\cctr@det}
1313 \ADD{\cctr@sol}{\cctr@det}{\cctr@sol}
1314 \DETERMINANT(#4,#5;#7,#8){\cctr@det}\MULTIPLY{#3}{\cctr@det}{\cctr@det}
1315 \ADD{\cctr@sol}{\cctr@det}{\cctr@sol}
1316 \@NUMBERSOL}
1317
1318 \def\DETERMINANT(#1)#2{%

63
 1319 \begingroup
1320 \MATRIXSIZE(#1){\cctr@size}
1321 \ifnum\cctr@size=2
1322 \@@DETERMINANT(#1){#2}
1323 \else \@@@DETERMINANT(#1){#2}\fi\@OUTPUTSOL{#2}}

\INVERSEMATRIX Inverse of a matrix.

1324 \def\@@INVERSEMATRIX(#1,#2;#3,#4)(#5,#6;#7,#8){%
1325 \ifdim \cctr@@det\p@ <\cctr@epsilon % Matrix is singular
1326 \let#5\undefined
1327 \let#6\undefined
1328 \let#7\undefined
1329 \let#8\undefined
1330 \cctr@Warnsingmatrix{#1}{#2}{#3}{#4}%
1331 \else \COPY{#1}{#8}
1332 \COPY{#4}{#5}
1333 \MULTIPLY{-1}{#3}{#7}
1334 \MULTIPLY{-1}{#2}{#6}
1335 \DIVIDE{1}{\cctr@det}{\cctr@det}
1336 \SCALARMATRIXPRODUCT{\cctr@det}(#5,#6;#7,#8)(#5,#6;#7,#8)
1337 \fi}
1338
1339 \def\@@@INVERSEMATRIX(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1340 \ifdim \cctr@@det\p@ <\cctr@epsilon % Matrix is singular
1341 \@TDMATRIXNOSOL(\cctr@solAA,\cctr@solAB,\cctr@solAC;
1342 \cctr@solBA,\cctr@solBB,\cctr@solBC;
1343 \cctr@solCA,\cctr@solCB,\cctr@solCC)
1344 \cctr@WarnsingTDmatrix{#1}{#2}{#3}{#4}{#5}{#6}{#7}{#8}{#9}%
1345 \else
1346 \@ADJMATRIX(#1,#2,#3;#4,#5,#6;#7,#8,#9)
1347 \@SCLRDIVVECT{\cctr@det}(\cctr@solAA,\cctr@solAB,\cctr@solAC)(%
1348 \cctr@solAA,\cctr@solAB,\cctr@solAC)
1349 \@SCLRDIVVECT{\cctr@det}(\cctr@solBA,\cctr@solBB,\cctr@solBC)(%
1350 \cctr@solBA,\cctr@solBB,\cctr@solBC)
1351 \@SCLRDIVVECT{\cctr@det}(\cctr@solCA,\cctr@solCB,\cctr@solCC)(%
1352 \cctr@solCA,\cctr@solCB,\cctr@solCC)
1353 \fi
1354 \@@TDMATRIXSOL}
1355
1356 \def\@SCLRDIVVECT#1(#2,#3,#4)(#5,#6,#7){%
1357 \DIVIDE{#2}{#1}{#5}\DIVIDE{#3}{#1}{#6}\DIVIDE{#4}{#1}{#7}}
1358
1359 \def\@ADJMATRIX(#1,#2,#3;#4,#5,#6;#7,#8,#9){%
1360 \DETERMINANT(#5,#6;#8,#9){\cctr@solAA}
1361 \DETERMINANT(#6,#4;#9,#7){\cctr@solBA}
1362 \DETERMINANT(#4,#5;#7,#8){\cctr@solCA}
1363 \DETERMINANT(#8,#9;#2,#3){\cctr@solAB}
1364 \DETERMINANT(#1,#3;#7,#9){\cctr@solBB}
1365 \DETERMINANT(#2,#1;#8,#7){\cctr@solCB}
1366 \DETERMINANT(#2,#3;#5,#6){\cctr@solAC}

64
 1367 \DETERMINANT(#3,#1;#6,#4){\cctr@solBC}
1368 \DETERMINANT(#1,#2;#4,#5){\cctr@solCC}}
1369
1370 \def\INVERSEMATRIX(#1)(#2){%
1371 \begingroup
1372 \DETERMINANT(#1){\cctr@det}
1373 \ABSVALUE{\cctr@det}{\cctr@@det}
1374 \MATRIXSIZE(#1){\cctr@size}
1375 \ifnum\cctr@size=2
1376 \@@INVERSEMATRIX(#1)(#2)
1377 \else
1378 \@@@INVERSEMATRIX(#1)(#2)\fi\@OUTPUTMATRIX(#2)}

\SOLVELINEARSYSTEM Solving a linear system (two equations and two unknowns or three equations and three un-
knowns).
1379 \def\@INCSYS#1#2{\cctr@WarnIncLinSys
1380 \let#1\undefined\let#2\undefined}
1381
1382 \def\@SOLPART#1#2#3#4{\cctr@WarnIndLinSys
1383 \DIVIDE{#1}{#2}{#3}
1384 \COPY{0}{#4}}
1385
1386 \def\@TDINCSYS(#1,#2,#3){\cctr@WarnIncTDLinSys
1387 \let#1\undefined
1388 \let#2\undefined
1389 \let#3\undefined}
1390
1391 \def\@@SOLVELINEARSYSTEM(#1,#2;#3,#4)(#5,#6)(#7,#8){%
1392 \DETERMINANT(#1,#2;#3,#4)\cctr@deta
1393 \DETERMINANT(#5,#2;#6,#4)\cctr@detb
1394 \DETERMINANT(#1,#5;#3,#6)\cctr@detc
1395 \ABSVALUE{\cctr@deta}{\cctr@@deta}
1396 \ABSVALUE{\cctr@detb}{\cctr@@detb}
1397 \ABSVALUE{\cctr@detc}{\cctr@@detc}
1398 \ifdim \cctr@@deta\p@>\cctr@epsilon% Regular matrix. Determinate system
1399 \DIVIDE{\cctr@detb}{\cctr@deta}{#7}
1400 \DIVIDE{\cctr@detc}{\cctr@deta}{#8}
1401 \else % Singular matrix \cctr@deta=0
1402 \ifdim \cctr@@detb\p@>\cctr@epsilon% Incompatible system
1403 \@INCSYS#7#8
1404 \else
1405 \ifdim \cctr@@detc\p@>\cctr@epsilon% Incompatible system
1406 \@INCSYS#7#8
1407 \else
1408 \MATRIXABSVALUE(#1,#2;#3,#4)(\cctr@tempa,\cctr@tempb;
1409 \cctr@tempc,\cctr@tempd)
1410 \ifdim \cctr@tempa\p@ > \cctr@epsilon
1411 % Indeterminate system
1412 \@SOLPART{#5}{#1}{#7}{#8}
1413 \else

65
 1414 \ifdim \cctr@tempb\p@ > \cctr@epsilon
1415 % Indeterminate system
1416 \@SOLPART{#5}{#2}{#8}{#7}
1417 \else
1418 \ifdim \cctr@tempc\p@ > \cctr@epsilon
1419 % Indeterminate system
1420 \@SOLPART{#6}{#3}{#7}{#8}
1421 \else
1422 \ifdim \cctr@tempd\p@ > \cctr@epsilon
1423 % Indeterminate system
1424 \@SOLPART{#6}{#4}{#8}{#7}
1425 \else
1426 \VECTORNORM(#5,#6){\cctr@tempa}
1427 \ifdim \cctr@tempa\p@ > \cctr@epsilon
1428 % Incompatible system
1429 \@INCSYS#7#8
1430 \else
1431 \cctr@WarnZeroLinSys
1432 \COPY{0}{#7}\COPY{0}{#8}
1433 % 0x=0 Indeterminate system
1434 \fi\fi\fi\fi\fi\fi\fi\fi}
1435
1436 \def\@@@SOLVELINEARSYSTEM(#1)(#2)(#3){%
1437 \DETERMINANT(#1){\cctr@det}
1438 \ABSVALUE{\cctr@det}{\cctr@@det}
1439 \ifdim\cctr@@det\p@<\cctr@epsilon
1440 \@TDINCSYS(#3)
1441 \else
1442 \@ADJMATRIX(#1)
1443 \MATRIXVECTORPRODUCT(\cctr@solAA,\cctr@solAB,\cctr@solAC;
1444 \cctr@solBA,\cctr@solBB,\cctr@solBC;
1445 \cctr@solCA,\cctr@solCB,\cctr@solCC)(#2)(#3)
1446 \@SCLRDIVVECT{\cctr@det}(#3)(#3)
1447 \fi}
1448
1449 \def\SOLVELINEARSYSTEM(#1)(#2)(#3){%
1450 \begingroup
1451 \MATRIXSIZE(#1){\cctr@size}
1452 \ifnum\cctr@size=2
1453 \@@SOLVELINEARSYSTEM(#1)(#2)(#3)
1454 \else
1455 \@@@SOLVELINEARSYSTEM(#1)(#2)(#3)
1456 \fi\@OUTPUTVECTOR(#3)}

Predefined numbers
\numberPI The number π
1457 \def\numberPI{3.14159}

\numberTWOPI 2π

66
 1458 \MULTIPLY{\numberPI}{2}{\numberTWOPI}

\numberHALFPI π/2
1459 \DIVIDE{\numberPI}{2}{\numberHALFPI}

\numberTHREEHALFPI 3π/2
1460 \MULTIPLY{\numberPI}{1.5}{\numberTHREEHALFPI}

\numberTHIRDPI π/3
1461 \DIVIDE{\numberPI}{3}{\numberTHIRDPI}

\numberQUARTERPI π/4
1462 \DIVIDE{\numberPI}{4}{\numberQUARTERPI}

\numberFIFTHPI π/5
1463 \DIVIDE{\numberPI}{5}{\numberFIFTHPI}

\numberSIXTHPI π/6
1464 \DIVIDE{\numberPI}{6}{\numberSIXTHPI}

\numberE The number e

1465 \def\numberE{2.71828}

\numberINVE 1/e
1466 \DIVIDE{1}{\numberE}{\numberINVE}

\numberETWO e2
1467 \SQUARE{\numberE}{\numberETWO}

\numberINVETWO 1/e2
1468 \SQUARE{\numberINVE}{\numberINVETWO}

\numberLOGTEN log 10
1469 \def\numberLOGTEN{2.30258}

\numberGOLD The golden ratio φ

1470 \def\numberGOLD{1.61803}

\numberINVGOLD 1/φ
1471 \def\numberINVGOLD{0.61803}
√
\numberSQRTTWO 2
1472 \def\numberSQRTTWO{1.41421}
√
\numberSQRTTHREE 3
1473 \def\numberSQRTTHREE{1.73205}
√
\numberSQRTFIVE 5
1474 \def\numberSQRTFIVE{2.23607}

67
\numberCOSXLV cos 45o (or cos π/4)
1475 \def\numberCOSXLV{0.70711}

\numberCOSXXX cos 30o (or cos π/6)

1476 \def\numberCOSXXX{0.86603}

1477 h/calculatori

14 calculus
1478 h∗calculusi
1479 \NeedsTeXFormat{LaTeX2e}
1480 \ProvidesPackage{calculus}[2014/02/20 v.2.0]

This package requires the calculator package.

1481 \RequirePackage{calculator}
14.1 Error and info messages
For scalar functions
Error message to be issued when you attempt to define, with \newfunction, an already defined
command:
1482 \def\ccls@ErrorFuncDef#1{%
1483 \PackageError{calculus}%
1484 {\noexpand#1 command already defined}
1485 {The \noexpand#1 control sequence is already defined\MessageBreak
1486 If you want to redefine the \noexpand#1 command as a
1487 function\MessageBreak
1488 please, use the \noexpand\renewfunction command}}
Error message to be issued when you attempt to redefine, with \renewfunction, an undefined
command:
1489 \def\ccls@ErrorFuncUnDef#1{%
1490 \PackageError{calculus}%
1491 {\noexpand#1 command undefined}
1492 {The \noexpand#1 control sequence is not currently defined\MessageBreak
1493 If you want to define the \noexpand#1 command as a function\MessageBreak
1494 please, use the \noexpand\newfunction command}}
Info message to be issued when \ensurefunction does not changes an already defined command:
1495 \def\ccls@InfoFuncEns#1{%
1496 \PackageInfo{calculus}%
1497 {\noexpand#1 command already defined\MessageBreak
1498 the \noexpand\ensurefunction command will not redefine it}}

For polar functions

1499 \def\ccls@ErrorPFuncDef#1{%
1500 \PackageError{calculus}%
1501 {\noexpand#1 command already defined}

68
 1502 {The \noexpand#1 control sequence is already defined\MessageBreak
1503 If you want to redefine the \noexpand#1
1504 command as a polar function\MessageBreak
1505 please, use the \noexpand\renewpolarfunction command}}
1506
1507 \def\ccls@ErrorPFuncUnDef#1{%
1508 \PackageError{calculus}%
1509 {\noexpand#1 command undefined}
1510 {The \noexpand#1 control sequence
1511 is not currently defined.\MessageBreak
1512 If you want to define the \noexpand#1 command as a polar
1513 function\MessageBreak
1514 please, use the \noexpand\newpolarfunction command}}
1515
1516 \def\ccls@InfoPFuncEns#1{%
1517 \PackageInfo{calculus}%
1518 {\noexpand#1 command already defined\MessageBreak
1519 the \noexpand\ensurepolarfunction command does not redefine it}}
For vector functions
1520 \def\ccls@ErrorVFuncDef#1{%
1521 \PackageError{calculus}%
1522 {\noexpand#1 command already defined}
1523 {The \noexpand#1 control sequence is already defined\MessageBreak
1524 If you want to redefine the \noexpand#1 command as a vector
1525 function\MessageBreak
1526 please, use the \noexpand\renewvectorfunction command}}
1527
1528 \def\ccls@ErrorVFuncUnDef#1{%
1529 \PackageError{calculus}%
1530 {\noexpand#1 command undefined}
1531 {The \noexpand#1 control sequence is not currently
1532 defined.\MessageBreak
1533 If you want to define the \noexpand#1 command as a vector
1534 function\MessageBreak
1535 please, use the \noexpand\newvectorfunction command}}
1536
1537 \def\ccls@InfoVFuncEns#1{%
1538 \PackageInfo{calculus}%
1539 {\noexpand#1 command already defined\MessageBreak
1540 the \noexpand\ensurevectorfunction command does not redefine it}}

14.2 New functions

New scalar functions
\newfunction The \newfunction{#1}{#2} instruction defines a new function called #1. #2 is the list of
instructions to calculate the function \y and his derivative \Dy from the \t variable.
1541 \def\newfunction#1#2{%
1542 \ifx #1\undefined
1543 \ccls@deffunction{#1}{#2}

69
 1544 \else
1545 \ccls@ErrorFuncDef{#1}
1546 \fi}

\renewfunction \renewfunction redefines #1, as a new function, if this command is already defined.
1547 \def\renewfunction#1#2{%
1548 \ifx #1\undefined
1549 \ccls@ErrorFuncUnDef{#1}
1550 \else
1551 \ccls@deffunction{#1}{#2}
1552 \fi}

\ensurefunction \ensurefunction defines the new function #1 (only if this macro is undefined).
1553 \def\ensurefunction#1#2{%
1554 \ifx #1\undefined\ccls@deffunction{#1}{#2}
1555 \else
1556 \ccls@InfoFuncEns{#1}
1557 \fi}

\forcefunction \forcefunction defines (if undefined) or redefines (if defined) the new function #1.
1558 \def\forcefunction#1#2{%
1559 \ccls@deffunction{#1}{#2}}

\ccls@deffunction The private \ccls@deffunction command makes the real work. The new functions will have
three arguments: ##1, a number, ##2, the value of the new function in that number, and
##3, the derivative.
1560 \def\ccls@deffunction#1#2{%
1561 \def#1##1##2##3{%
1562 \begingroup
1563 \def\t{##1}%
1564 #2
1565 \xdef##2{\y}%
1566 \xdef##3{\Dy}%
1567 \endgroup}\ignorespaces}

New polar functions

\newpolarfunction The \newpolarfunction{#1}{#2} instruction defines a new polar function called #1. #2 is the
list of instructions to calculate the radius \r and his derivative \Dr from the \t arc variable.
1568 \def\newpolarfunction#1#2{%
1569 \ifx #1\undefined
1570 \ccls@defpolarfunction{#1}{#2}
1571 \else
1572 \ccls@ErrorPFuncDef{#1}
1573 \fi}

\renewpolarfunction \renewpolarfunction redefines #1 if already defined.

1574 \def\renewpolarfunction#1#2{%
1575 \ifx #1\undefined

70
 1576 \ccls@ErrorPFuncUnDef{#1}
1577 \else
1578 \ccls@defpolarfunction{#1}{#2}
1579 \fi}

\ensurepolarfunction \ensurepolarfunction defines (only if undefined) #1.

1580 \def\ensurepolarfunction#1#2{%
1581 \ifx #1\undefined\ccls@defpolarfunction{#1}{#2}
1582 \else
1583 \ccls@InfoPFuncEns{#1}
1584 \fi}

\forcepolarfunction \forcepolarfunction defines (if undefined) or redefines (if defined) #1.

1585 \def\forcepolarfunction#1#2{%
1586 \ccls@defpolarfunction{#1}{#2}}

\ccls@defpolarfunction The private \ccls@defpolarfunction command makes the real work. The new functions will
have three arguments: ##1, a number (the polar radius), ##2, ##3, ##4, and ##5, the x
and y component functions and its derivatives at ##1.
1587 \def\ccls@defpolarfunction#1#2{%
1588 \def#1##1##2##3##4##5{%
1589 \begingroup
1590 \def\t{##1}
1591 #2
1592 \COS{\t}\ccls@cost
1593 \MULTIPLY\r\ccls@cost{\x}
1594 \SIN{\t}\ccls@sint
1595 \MULTIPLY\r\ccls@sint{\y}
1596 \MULTIPLY\ccls@cost\Dr\Dx
1597 \SUBTRACT{\Dx}{\y}{\Dx}
1598 \MULTIPLY\ccls@sint\Dr\Dy
1599 \ADD{\Dy}{\x}{\Dy}
1600 \xdef##2{\x}
1601 \xdef##3{\Dx}
1602 \xdef##4{\y}
1603 \xdef##5{\Dy}
1604 \endgroup}\ignorespaces}

New vector functions

\newvectorfunction The \newvectorfunction{#1}{#2} instruction defines a new vector (parametric) function called
#1. #2 is the list of instructions to calculate \x, \y, \Dx and \Dy from the \t arc variable.
1605 \def\newvectorfunction#1#2{%
1606 \ifx #1\undefined
1607 \ccls@defvectorfunction{#1}{#2}
1608 \else
1609 \ccls@ErrorVFuncDef{#1}
1610 \fi}

71
 \renewvectorfunction \renewvectorfunction redefines #1 if already defined.
1611 \def\renewvectorfunction#1#2{%
1612 \ifx #1\undefined
1613 \ccls@ErrorVFuncUnDef{#1}
1614 \else
1615 \ccls@defvectorfunction{#1}{#2}
1616 \fi}

\ensurevectorfunction \ensurevectorfunction defines (only if undefined) #1.

1617 \def\ensurevectorfunction#1#2{%
1618 \ifx #1\undefined\ccls@defvectorfunction{#1}{#2}
1619 \else
1620 \ccls@InfoVFuncEns{#1}
1621 \fi}

\forcevectorfunction \forcevectorfunction defines (if undefined) or redefines (if defined) #1.

1622 \def\forcevectorfunction#1#2{%
1623 \ccls@defvectorfunction{#1}{#2}}

\ccls@defvectorfunction The private \ccls@defvectorfunction command makes the real work. The new functions will
have three arguments: ##1, a number, ##2, ##3, ##4, and ##5, the x and y component
functions and its derivatives at ##1.
1624 \def\ccls@defvectorfunction#1#2{%
1625 \def#1##1##2##3##4##5{%
1626 \begingroup
1627 \def\t{##1}
1628 #2
1629 \xdef##2{\x}
1630 \xdef##3{\Dx}
1631 \xdef##4{\y}
1632 \xdef##5{\Dy}
1633 \endgroup}\ignorespaces}

14.3 Polynomials
Linear (first degreee) polynomials
\newlpoly The \newlpoly{#1}{#2}{#3} instruction defines the linear polynomial
#1 = #2 + #3t.
1634 \def\newlpoly#1#2#3{%
1635 \newfunction{#1}{%
1636 \ccls@lpoly{#2}{#3}}}

\renewlpoly We define also the \renewlpoly, \ensurelpoly and \forcelpoly variants.

1637 \def\renewlpoly#1#2#3{%
1638 \renewfunction{#1}{%
1639 \ccls@lpoly{#2}{#3}}}

72
\ensurelpoly
1640 \def\ensurelpoly#1#2#3{%
1641 \ensurefunction{#1}{%
1642 \ccls@lpoly{#2}{#3}}}

\forcelpoly
1643 \def\forcelpoly#1#2#3{%
1644 \forcefunction{#1}{%
1645 \ccls@lpoly{#2}{#3}}}

\ccls@lpoly The \ccls@lpoly{#1}{#2} macro defines the new polynomial function.

1646 \def\ccls@lpoly#1#2{%
1647 \MULTIPLY{#2}{\t}{\y}
1648 \ADD{\y}{#1}{\y}
1649 \COPY{#2}{\Dy}}

Quadratic polynomials
\newqpoly The \newqpoly{#1}{#2}{#3}{#4} instruction defines the quadratic polynomial
#1 = #2 + #3t + #4t2 .
1650 \def\newqpoly#1#2#3#4{%
1651 \newfunction{#1}{%
1652 \ccls@qpoly{#2}{#3}{#4}}}

\renewqpoly
1653 \def\renewqpoly#1#2#3#4{%
1654 \renewfunction{#1}{%
1655 \ccls@qpoly{#2}{#3}{#4}}}

\ensureqpoly
1656 \def\ensureqpoly#1#2#3#4{%
1657 \ensurefunction{#1}{%
1658 \ccls@qpoly{#2}{#3}{#4}}}

\forceqpoly
1659 \def\forceqpoly#1#2#3#4{%
1660 \forcefunction{#1}{%
1661 \ccls@qpoly{#2}{#3}{#4}}}

\ccls@qpoly The \ccls@qpoly{#1}{#2} macro defines the new polynomial function.

1662 \def\ccls@qpoly#1#2#3{%
1663 \MULTIPLY{\t}{#3}{\y}
1664 \MULTIPLY{2}{\y}{\Dy}
1665 \ADD{#2}{\Dy}{\Dy}
1666 \ADD{#2}{\y}{\y}
1667 \MULTIPLY{\t}{\y}{\y}
1668 \ADD{#1}{\y}{\y}}

73
 Cubic polynomials
\newcpoly The \newcpoly{#1}{#2}{#3}{#4}{#5} instruction defines the cubic polynomial
#1 = #2 + #3t + #4t2 + #5t3 .
1669 \def\newcpoly#1#2#3#4#5{%
1670 \newfunction{#1}{%
1671 \ccls@cpoly{#2}{#3}{#4}{#5}}}

\renewcpoly
1672 \def\renewcpoly#1#2#3#4#5{%
1673 \renewfunction{#1}{%
1674 \ccls@cpoly{#2}{#3}{#4}{#5}}}

\ensurecpoly
1675 \def\ensurecpoly#1#2#3#4#5{%
1676 \ensurefunction{#1}{%
1677 \ccls@cpoly{#2}{#3}{#4}{#5}}}

\forcecpoly
1678 \def\forcecpoly#1#2#3#4#5{%
1679 \forcefunction{#1}{%
1680 \ccls@cpoly{#2}{#3}{#4}{#5}}}

\ccls@cpoly The \ccls@cpoly{#1}{#2} macro defines the new polynomial function.

1681 \def\ccls@cpoly#1#2#3#4{%
1682 \MULTIPLY{\t}{#4}{\y}
1683 \MULTIPLY{3}{\y}{\Dy}
1684 \ADD{#3}{\y}{\y}
1685 \MULTIPLY{2}{#3}{\ccls@temp}
1686 \ADD{\ccls@temp}{\Dy}{\Dy}
1687 \MULTIPLY{\t}{\y}{\y}
1688 \MULTIPLY{\t}{\Dy}{\Dy}
1689 \ADD{#2}{\y}{\y}
1690 \ADD{#2}{\Dy}{\Dy}
1691 \MULTIPLY{\t}{\y}{\y}
1692 \ADD{#1}{\y}{\y}
1693 }

14.4 Elementary functions

\ONEfunction The \ONEfunction: y(t) = 1, y 0 (t) = 0
1694 \newfunction{\ONEfunction}{%
1695 \COPY{1}{\y}
1696 \COPY{0}{\Dy}}

\ZEROfunction The \ZEROfunction: y(t) = 0, y 0 (t) = 0

1697 \newfunction{\ZEROfunction}{%
1698 \COPY{0}{\y}
1699 \COPY{0}{\Dy}}

74
 \IDENTITYfunction The \IDENTITYfunction: y(t) = t, y 0 (t) = 1
1700 \newfunction{\IDENTITYfunction}{%
1701 \COPY{\t}{\y}
1702 \COPY{1}{\Dy}}

\RECIPROCALfunction The \RECIPROCALfunction: y(t) = 1/t, y 0 (t) = −1/t2

1703 \newfunction{\RECIPROCALfunction}{%
1704 \DIVIDE{1}{\t}{\y}
1705 \SQUARE{\y}{\Dy}
1706 \MULTIPLY{-1}{\Dy}{\Dy}}

\SQUAREfunction The \SQUAREfunction: y(t) = t2 , y 0 (t) = 2t

1707 \newfunction{\SQUAREfunction}{%
1708 \SQUARE{\t}{\y}
1709 \MULTIPLY{2}{\t}{\Dy}}

\CUBEfunction The \CUBEfunction: y(t) = t3 , y 0 (t) = 3t2

1710 \newfunction{\CUBEfunction}{%
1711 \SQUARE{\t}{\Dy}
1712 \MULTIPLY{\t}{\Dy}{\y}
1713 \MULTIPLY{3}{\Dy}{\Dy}}
√ 0 √
\SQRTfunction The \SQRTfunction: y(t) = t, y (t) = 1/(2 t)
1714 \newfunction{\SQRTfunction}{%
1715 \SQRT{\t}{\y}
1716 \DIVIDE{0.5}{\y}{\Dy}}

\EXPfunction The \EXPfunction: y(t) = exp t, y 0 (t) = exp t

1717 \newfunction{\EXPfunction}{%
1718 \EXP{\t}{\y}
1719 \COPY{\y}{\Dy}}

\COSfunction The \COSfunction: y(t) = cos t, y 0 (t) = − sin t

1720 \newfunction{\COSfunction}{%
1721 \COS{\t}{\y}
1722 \SIN{\t}{\Dy}
1723 \MULTIPLY{-1}{\Dy}{\Dy}}

\SINfunction The \SINfunction: y(t) = sin t, y 0 (t) = cos t

1724 \newfunction{\SINfunction}{%
1725 \SIN{\t}{\y}
1726 \COS{\t}{\Dy}}

\TANfunction The \TANfunction: y(t) = tan t, y 0 (t) = 1/(cos t)2

1727 \newfunction{\TANfunction}{%
1728 \TAN{\t}{\y}
1729 \COS{\t}{\Dy}
1730 \SQUARE{\Dy}{\Dy}
1731 \DIVIDE{1}{\Dy}{\Dy}}

75
 \COTfunction The \COTfunction: y(t) = cot t, y 0 (t) = −1/(sin t)2
1732 \newfunction{\COTfunction}{%
1733 \COTAN{\t}{\y}
1734 \SIN{\t}{\Dy}
1735 \SQUARE{\Dy}{\Dy}
1736 \DIVIDE{-1}{\Dy}{\Dy}}

\COSHfunction The \COSHfunction: y(t) = cosh t, y 0 (t) = sinh t

1737 \newfunction{\COSHfunction}{%
1738 \COSH{\t}{\y}
1739 \SINH{\t}{\Dy}}

\SINHfunction The \SINHfunction: y(t) = sinh t, y 0 (t) = cosh t

1740 \newfunction{\SINHfunction}{%
1741 \SINH{\t}{\y}
1742 \COSH{\t}{\Dy}}

\TANHfunction The \TANHfunction: y(t) = tanh t, y 0 (t) = 1/(cosh t)2

1743 \newfunction{\TANHfunction}{%
1744 \TANH{\t}{\y}
1745 \COSH{\t}{\Dy}
1746 \SQUARE{\Dy}{\Dy}
1747 \DIVIDE{1}{\Dy}{\Dy}}

\COTHfunction The \COTHfunction: y(t) = coth t, y 0 (t) = −1/(sinh t)2

1748 \newfunction{\COTHfunction}{%
1749 \COTANH{\t}{\y}
1750 \SINH{\t}{\Dy}
1751 \SQUARE{\Dy}{\Dy}
1752 \DIVIDE{-1}{\Dy}{\Dy}}

\LOGfunction The \LOGfunction: y(t) = log t, y 0 (t) = 1/t

1753 \newfunction{\LOGfunction}{%
1754 \LOG{\t}{\y}
1755 \DIVIDE{1}{\t}{\Dy}}
(
0 if t < 0 0
\HEAVISIDEfunction The \HEAVISIDEfunction: y(t) = , y (t) = 0
1 if t ≥ 0
1756 \newfunction{\HEAVISIDEfunction}{%
1757 \ifdim \t\p@<\z@ \COPY{0}{\y}\else\COPY{1}{\y}\fi
1758 \COPY{0}{\Dy}}

√
\ARCSINfunction The \ARCSINfunction: y(t) = arcsin t, y 0 (t) = 1/ 1 − t2
1759 \newfunction{\ARCSINfunction}{%
1760 \ARCSIN{\t}{\y}
1761 \SQUARE{\t}{\yy}
1762 \SUBTRACT{1}{\yy}{\yy}
1763 \SQRT{\yy}{\Dy}
1764 \DIVIDE{1}{\Dy}{\Dy}}

76
 √
\ARCCOSfunction The \ARCCOSfunction: y(t) = arccos t, y 0 (t) = −1/ 1 − t2
1765 \newfunction{\ARCCOSfunction}{%
1766 \ARCCOS{\t}{\y}
1767 \SQUARE{\t}{\yy}
1768 \SUBTRACT{1}{\yy}{\yy}
1769 \SQRT{\yy}{\Dy}
1770 \DIVIDE{-1}{\Dy}{\Dy}}

\ARCTANfunction The \ARCTANfunction: y(t) = arctan t, y 0 (t) = 1/(1 + t2 )

1771 \newfunction{\ARCTANfunction}{%
1772 \ARCTAN{\t}{\y}
1773 \SQUARE{\t}{\yy}
1774 \ADD{1}{\yy}{\yy}
1775 \DIVIDE{1}{\yy}{\Dy}}

\ARCCOTfunction The \ARCCOTfunction: y(t) = arccot t, y 0 (t) = −1/(1 + t2 )

1776 \newfunction{\ARCCOTfunction}{%
1777 \ARCCOT{\t}{\y}
1778 \SQUARE{\t}{\yy}
1779 \ADD{1}{\yy}{\yy}
1780 \DIVIDE{-1}{\yy}{\Dy}}
√
\ARSINHfunction The \ARSINHfunction: y(t) = arsinh t, y 0 (t) = 1/ 1 + t2
1781 \newfunction{\ARSINHfunction}{%
1782 \ARSINH{\t}{\y}
1783 \SQUARE{\t}{\yy}
1784 \ADD{1}{\yy}{\yy}
1785 \SQRT{\yy}{\Dy}
1786 \DIVIDE{1}{\Dy}{\Dy}}
√
\ARCOSHfunction The \ARSINHfunction: y(t) = arcosh t, y 0 (t) = 1/ t2 − 1
1787 \newfunction{\ARCOSHfunction}{%
1788 \ARCOSH{\t}{\y}
1789 \SQUARE{\t}{\yy}
1790 \SUBTRACT{\yy}{1}{\yy}
1791 \SQRT{\yy}{\Dy}
1792 \DIVIDE{1}{\Dy}{\Dy}}

\ARTANHfunction The \ARTANHfunction: y(t) = artanh t, y 0 (t) = 1/(t2 − 1)

1793 \newfunction{\ARTANHfunction}{%
1794 \ARTANH{\t}{\y}
1795 \SQUARE{\t}{\yy}
1796 \SUBTRACT{1}{\yy}{\yy}
1797 \DIVIDE{1}{\yy}{\Dy}}

\ARCOTHfunction The \ARCOTHfunction: y(t) = arcoth t, y 0 (t) = 1/(t2 − 1)

1798 \newfunction{\ARCOTHfunction}{%
1799 \ARCOTH{\t}{\y}
1800 \SQUARE{\t}{\yy}
1801 \SUBTRACT{1}{\yy}{\yy}
1802 \DIVIDE{1}{\yy}{\Dy}}

77
 14.5 Operations with functions
\CONSTANTfunction \CONSTANTfunction defines #2 as the constant function f (t) = #1.
1803 \def\CONSTANTfunction#1#2{%
1804 \def#2##1##2##3{%
1805 \xdef##2{#1}%
1806 \xdef##3{0}}}

\SUMfunction \SUMfunction defines #3 as the sum of functions #1 and #2.

1807 \def\SUMfunction#1#2#3{%
1808 \def#3##1##2##3{%
1809 \begingroup
1810 #1{##1}{\ccls@SUMf}{\ccls@SUMDf}%
1811 #2{##1}{\ccls@SUMg}{\ccls@SUMDg}%
1812 \ADD{\ccls@SUMf}{\ccls@SUMg}{\ccls@SUMfg}
1813 \ADD{\ccls@SUMDf}{\ccls@SUMDg}{\ccls@SUMDfg}
1814 \xdef##2{\ccls@SUMfg}%
1815 \xdef##3{\ccls@SUMDfg}%
1816 \endgroup}\ignorespaces}

\SUBTRACTfunction \SUBTRACTfunction defines #3 as the difference of functions #1 and #2.

1817 \def\SUBTRACTfunction#1#2#3{%
1818 \def#3##1##2##3{%
1819 \begingroup
1820 #1{##1}{\ccls@SUBf}{\ccls@SUBDf}%
1821 #2{##1}{\ccls@SUBg}{\ccls@SUBDg}%
1822 \SUBTRACT{\ccls@SUBf}{\ccls@SUBg}{\ccls@SUBfg}
1823 \SUBTRACT{\ccls@SUBDf}{\ccls@SUBDg}{\ccls@SUBDfg}
1824 \xdef##2{\ccls@SUBfg}%
1825 \xdef##3{\ccls@SUBDfg}%
1826 \endgroup}\ignorespaces}

\PRODUCTfunction \PRODUCTfunction defines #3 as the product of functions #1 and #2.

1827 \def\PRODUCTfunction#1#2#3{%
1828 \def#3##1##2##3{%
1829 \begingroup
1830 #1{##1}{\ccls@PROf}{\ccls@PRODf}%
1831 #2{##1}{\ccls@PROg}{\ccls@PRODg}%
1832 \MULTIPLY{\ccls@PROf}{\ccls@PROg}{\ccls@PROfg}
1833 \MULTIPLY{\ccls@PROf}{\ccls@PRODg}{\ccls@PROfDg}
1834 \MULTIPLY{\ccls@PRODf}{\ccls@PROg}{\ccls@PRODfg}
1835 \ADD{\ccls@PROfDg}{\ccls@PRODfg}{\ccls@PRODfg}
1836 \xdef##2{\ccls@PROfg}%
1837 \xdef##3{\ccls@PRODfg}%
1838 \endgroup}\ignorespaces}

\QUOTIENTfunction \QUOTIENTfunction defines #3 as the quotient of functions #1 and #2.

1839 \def\QUOTIENTfunction#1#2#3{%
1840 \def#3##1##2##3{%
1841 \begingroup

78
 1842 #1{##1}{\ccls@QUOf}{\ccls@QUODf}%
1843 #2{##1}{\ccls@QUOg}{\ccls@QUODg}%
1844 \DIVIDE{\ccls@QUOf}{\ccls@QUOg}{\ccls@QUOfg}
1845 \MULTIPLY{\ccls@QUOf}{\ccls@QUODg}{\ccls@QUOfDg}
1846 \MULTIPLY{\ccls@QUODf}{\ccls@QUOg}{\ccls@QUODfg}
1847 \SUBTRACT{\ccls@QUODfg}{\ccls@QUOfDg}{\ccls@QUOnum}
1848 \SQUARE{\ccls@QUOg}{\ccls@qsquaretempg}
1849 \DIVIDE{\ccls@QUOnum}{\ccls@qsquaretempg}{\ccls@QUODfg}
1850 \xdef##2{\ccls@QUOfg}%
1851 \xdef##3{\ccls@QUODfg}%
1852 \endgroup}\ignorespaces}

\COMPOSITIONfunction \COMPOSITIONfunction defines #3 as the composition of functions #1 and #2.

1853 \def\COMPOSITIONfunction#1#2#3{% #3=#1(#2)
1854 \def#3##1##2##3{%
1855 \begingroup
1856 #2{##1}{\ccls@COMg}{\ccls@COMDg}%
1857 #1{\ccls@COMg}{\ccls@COMf}{\ccls@COMDf}%
1858 \MULTIPLY{\ccls@COMDg}{\ccls@COMDf}{\ccls@COMDf}
1859 \xdef##2{\ccls@COMf}%
1860 \xdef##3{\ccls@COMDf}%
1861 \endgroup}\ignorespaces}

\SCALEfunction \SCALEfunction defines #3 as the product of number #1 and function #2.

1862 \def\SCALEfunction#1#2#3{%
1863 \def#3##1##2##3{%
1864 \begingroup
1865 #2{##1}{\ccls@SCFf}{\ccls@SCFDf}%
1866 \MULTIPLY{#1}{\ccls@SCFf}{\ccls@SCFaf}
1867 \MULTIPLY{#1}{\ccls@SCFDf}{\ccls@SCFDaf}
1868 \xdef##2{\ccls@SCFaf}%
1869 \xdef##3{\ccls@SCFDaf}%
1870 \endgroup}\ignorespaces}

\SCALEVARIABLEfunction \SCALEVARIABLEfunction scales the variable by number #1 and aplies function #2.
1871 \def\SCALEVARIABLEfunction#1#2#3{%
1872 \def#3##1##2##3{%
1873 \begingroup%
1874 \MULTIPLY{#1}{##1}{\ccls@SCVat}
1875 #2{\ccls@SCVat}{\ccls@SCVf}{\ccls@SCVDf}%
1876 \MULTIPLY{#1}{\ccls@SCVDf}{\ccls@SCVDf}
1877 \xdef##2{\ccls@SCVf}%
1878 \xdef##3{\ccls@SCVDf}%
1879 \endgroup}\ignorespaces}

\POWERfunction \POWERfunction defines #3 as the power of function #1 to exponent #2.

1880 \def\POWERfunction#1#2#3{%
1881 \def#3##1##2##3{%
1882 \begingroup
1883 #1{##1}{\ccls@POWf}{\ccls@POWDf}%

79
 1884 \POWER{\ccls@POWf}{#2}{\ccls@POWfn}
1885 \SUBTRACT{#2}{1}{\ccls@nminusone}
1886 \POWER{\ccls@POWf}{\ccls@nminusone}{\ccls@POWDfn}
1887 \MULTIPLY{#2}{\ccls@POWDfn}{\ccls@POWDfn}
1888 \MULTIPLY{\ccls@POWDfn}{\ccls@POWDf}{\ccls@POWDfn}
1889 \xdef##2{\ccls@POWfn}%
1890 \xdef##3{\ccls@POWDfn}%
1891 \endgroup}\ignorespaces}

LINEARCOMBINATIONfunction \LINEARCOMBINATIONfunction defines the new function #5 as the linear combination #1#2+#3#4.
#1 and #3 are two numbers. #1 and #3 are two functions.
1892 \def\LINEARCOMBINATIONfunction#1#2#3#4#5{%
1893 \def#5##1##2##3{%
1894 \begingroup
1895 #2{##1}{\ccls@LINf}{\ccls@LINDf}%
1896 #4{##1}{\ccls@LINg}{\ccls@LINDg}%
1897 \MULTIPLY{#1}{\ccls@LINf}{\ccls@LINf}
1898 \MULTIPLY{#3}{\ccls@LINg}{\ccls@LINg}
1899 \MULTIPLY{#1}{\ccls@LINDf}{\ccls@LINDf}
1900 \MULTIPLY{#3}{\ccls@LINDg}{\ccls@LINDg}
1901 \ADD{\ccls@LINf}{\ccls@LINg}{\ccls@LINafbg}
1902 \ADD{\ccls@LINDf}{\ccls@LINDg}{\ccls@LINDafbg}
1903 \xdef##2{\ccls@LINafbg}%
1904 \xdef##3{\ccls@LINDafbg}%
1905 \endgroup}\ignorespaces}

\POLARfunction \POLARfunction defines the polar curve #2. #1 is a previously defined function.
1906 \def\POLARfunction#1#2{%
1907 \PRODUCTfunction{#1}{\COSfunction}{\ccls@polarx}
1908 \PRODUCTfunction{#1}{\SINfunction}{\ccls@polary}
1909 \PARAMETRICfunction{\ccls@polarx}{\ccls@polary}{#2}}

\PARAMETRICfunction \PARAMETRICfunction defines the parametric curve #3. #1 and #2 are the components func-
tions (two previuosly defined functions).
1910 \def\PARAMETRICfunction#1#2#3{%
1911 \def#3##1##2##3##4##5{%
1912 #1{##1}{##2}{##3}
1913 #2{##1}{##4}{##5}}}

\VECTORfunction \VECTORfunction: an alias of \PARAMETRICfunction.

1914 \let\VECTORfunction\PARAMETRICfunction

1915 % </calculus>

80
Change History

v1.0 \ARCOTHfunction . . . . . . . . . . . . . . 77
General: First public version . . . . . . . . . . 1 New commands: \ARSINH, \ARCOSH,
v1.0a \ARTANH, \ARCOTH . . . . . . . . . . . . . . 53
General: calculator.dtx modified to make New commands: \DOTPRODUCT,
it autoinstallable. calculus.dtx \VECTORPRODUCT, \CROSSPRODUCT . . . 56
embedded in calculus.dtx . . . . . . . . . 1 New commands: \LENGTHADD,
v2.0 \LENGTHSUBTRACT . . . . . . . . . . . . . . 34
General: new calculator.dtx and Trivial error in documentation corrected 68
calculator.ins files . . . . . . . . . . . . . . . 1
v2.1
New commands: \ARCSINfunction,
\ARCCOSfunction, \ARCTANfunction, \@BASICLOG: Changed stop criterion on
\ARCCOTfunction . . . . . . . . . . . . . . 76 iterations to 2sp . . . . . . . . . . . . . . . 49
New commands: \ARCSIN, \ARCCOS, \FRACTIONALPART: Bug fixed . . . . . . . . . 36
\ARCTAN, \ARCCOT . . . . . . . . . . . . . . 49 \ROUND: Bug fixed . . . . . . . . . . . . . . . . . 37
New commands: \ARSINHfunction, \TRUNCATE: Bug fixed . . . . . . . . . . . . . . . 36
\ARCOSHfunction, \ARTANHfunction, General: Some bugs fixed . . . . . . . . . . . . . 1

Index
Numbers written in italic refer to the page where the corresponding entry is described; numbers
underlined refer to the code line of the definition; numbers in roman refer to the code lines where
the entry is used.

Symbols \@@@TRANSPOSEMATRIX . . . . \@@MATRIXCOPY . . . 1142, 1152

\# . . . . . . . . . . . . . . . . . 737 . . . . . . . . . . 1187, 1196 \@@MATRIXGLOBALCOPY . . . .
\@@@DETERMINANT . 1310, 1323 \@@@TRUNCATE . . 293, 295, 299 . . . . . . . . . . 1154, 1164
\@@@INVERSEMATRIX 1339, 1378 \@@@VECTORABSVALUE 1044, 1051 \@@MATRIXPRODUCT 1289, 1303
\@@@MATRIXABSVALUE 1232, 1240 \@@@VECTORADD . . . 1020, 1029 \@@MATRIXSUB . . . . 1213, 1227
\@@@MATRIXADD . . . 1201, 1212 \@@@VECTORCOPY . . . . 949, 956 \@@MATRIXVECTORPRODUCT .
\@@@MATRIXCOPY . . 1145, 1153 \@@@VECTORGLOBALCOPY 960, 967 . . . . . . . . . . 1241, 1254
\@@@MATRIXGLOBALCOPY . . . \@@@VECTORMATRIXPRODUCT \@@OUTPUTMATRIX . 1166, 1182
. . . . . . . . . . 1157, 1165 . . . . . . . . . . 1260, 1272 \@@OUTPUTVECTOR . . . 968, 978
\@@@MATRIXPRODUCT 1293, 1304 \@@@VECTORPRODUCT 1004, 1014 \@@ROUND . . . . . . . . . 307–309
\@@@MATRIXSUB . . . 1217, 1228 \@@@VECTORSUB . . . 1033, 1040 \@@SCALARMATRIXPRODUCT .
\@@@MATRIXVECTORPRODUCT \@@DEGREESCOS . . . . . 536, 601 . . . . . . . . . . 1273, 1287
. . . . . . . . . . 1245, 1255 \@@DEGREESCOT . . . . . 538, 605 \@@SCALARPRODUCT . . 980, 996
\@@@OUTPUTMATRIX 1170, 1183 \@@DEGREESSIN . . . . . 535, 599 \@@SCALARVECTORPRODUCT .
\@@@OUTPUTVECTOR . . 972, 979 \@@DEGREESTAN . . . . . 537, 603 . . . . . . . . . . 1052, 1064
\@@@ROUND . . . . 315, 317, 322 \@@DETERMINANT . . 1305, 1322 \@@SOLVELINEARSYSTEM . . .
\@@@SCALARMATRIXPRODUCT \@@EXP . . . . . . . . . . . 609, 610 . . . . . . . . . . 1391, 1453
. . . . . . . . . . 1277, 1288 \@@FRACTIONALPART . 274, 276 \@@TDMATRIXSOL . . 1133, 1354
\@@@SCALARPRODUCT . 985, 997 \@@INTEGERPART . . . . 259, 261 \@@TRANSPOSEMATRIX 1184, 1195
\@@@SCALARVECTORPRODUCT \@@INVERSEMATRIX 1324, 1376 \@@TRUNCATE . . . . . . . 285–287
. . . . . . . . . . 1056, 1065 \@@LOG . . . . . . . . . . . 713, 732 \@@VECTORABSVALUE 1041, 1050
\@@@SOLVELINEARSYSTEM . . \@@MATRIXABSVALUE 1229, 1239 \@@VECTORADD . . . . 1016, 1028
. . . . . . . . . . 1436, 1455 \@@MATRIXADD . . . . 1197, 1211 \@@VECTORCOPY . . . . . 946, 955

81
\@@VECTORGLOBALCOPY 957, 966 \@SOLPART . . . . . . . 1382, \ARCOTHfunction ..... 1798
\@@VECTORMATRIXPRODUCT . 1412, 1416, 1420, 1424 \ARCSIN . . . . . 754, 829, 1760
. . . . . . . . . . 1256, 1271 \@TDINCSYS . . . . . 1386, 1440 \ARCSINfunction ..... 1759
\@@VECTORPRODUCT . 999, 1013 \@TDMATRIXCOPY . . . . . . . . \ARCTAN . . . . . 832, 878, 1772
\@@VECTORSUB . . . . 1030, 1039 . 1092, 1146, 1158, 1188 \ARCTANfunction ..... 1771
\@ADJMATRIX . 1346, 1359, 1442 \@TDMATRIXGLOBALSOL . . . . \ARSINH . . . . . . . . . 881, 1782
\@BASICARCTAN . . . . . 844, 847 . . . . . . . . . . 1112, 1159 \ARSINHfunction ..... 1781
\@BASICEXP . . . 629, 637, 644 \@TDMATRIXNOSOL . . . . . . . \ARTANH . . . . . . . . . 901, 1794
\@BASICLOG . . . . . . . 730, 736 . . . . . 1122, 1135, 1341 \ARTANHfunction ..... 1793
\@BASICSINE . . . 422, 431, 553 \@TDMATRIXSOL . . . . 1102,
\@BASICTAN . . . 468, 473, 581 1137, 1147, 1189, C
\@CONVERTDEG . . . . . . . . . . 1205, 1221, 1281, 1297 \ccls@COMDf . 1857, 1858, 1860
. 599, 601, 603, 605, 607 \@TRUNCATE . . . . . . . 285, 286 \ccls@COMDg . . . . . 1856, 1858
\@DEGREES . . . . . . . . . . . . \@VECTORSIZE . . . . . . 943, 944 \ccls@COMf . . . . . 1857, 1859
. 600, 602, 604, 606–608 \ccls@COMg . . . . . 1856, 1857
\@DEGREESCOS . . . . . . 536, 562 A \ccls@cost . 1592, 1593, 1596
\@DEGREESCOT . . . . . . 538, 586 \ABSVALUE . . . . . 154, 167, \ccls@cpoly . . . . . . 1671,
\@DEGREESSIN . . 535, 539, 600 168, 242, 349, 350, 1674, 1677, 1680, 1681
\@DEGREESTAN . . . . . . 537, 566 374, 433, 475, 622, \ccls@deffunction 1543,
\@DIVIDE . . 174, 184, 189, 243 667, 680, 693, 704, 1551, 1554, 1559, 1560
\@EXP . . . . . . . . 609, 618, 620 790, 805, 1042, 1045, \ccls@defpolarfunction .
\@FRACTIONALPART . . 276, 282 1373, 1395–1397, 1438 . . . . . . . . . . . 1570,
\@INCSYS 1379, 1403, 1406, 1429 \ADD . . . . . . . . . . . . . 160, 1578, 1581, 1586, 1587
\@INTEGERDIVIDE . . . 190, 193 164, 186, 238, 247, \ccls@defvectorfunction
\@INTEGERPART . . . . . 261, 271 251, 260, 275, 324– . . . . . . . . . . . 1607,
\@LOG . . . . . 713, 714, 733, 734 326, 329–331, 334– 1615, 1618, 1623, 1624
\@MATRIXSIZE . . . . 1139, 1140 337, 340–344, 399, \ccls@ErrorFuncDef 1482, 1545
\@NUMBERSOL . . . . . 1138, 1316 418, 451, 464, 516, \ccls@ErrorFuncUnDef . . .
\@OUTPUTMATRIX . . . 1166, 527, 548, 564, 577, . . . . . . . . . . 1489, 1549
1196, 1212, 1228, 648, 651, 654, 657, \ccls@ErrorPFuncDef . . . .
1240, 1288, 1304, 1378 660, 664, 675, 723, . . . . . . . . . . 1499, 1572
\@OUTPUTSOL 144, 188, 216, 746, 782, 853, 856, \ccls@ErrorPFuncUnDef . .
226, 239, 255, 258, 859, 862, 865, 868, . . . . . . . . . . 1507, 1576
272, 283, 297, 320, 871, 873, 884, 886, \ccls@ErrorVFuncDef . . . .
367, 407, 430, 448, 898, 912, 929, 983, . . . . . . . . . . 1520, 1609
452, 472, 497, 510, 988, 990, 1017, 1018, \ccls@ErrorVFuncUnDef . .
561, 565, 585, 598, 1021–1023, 1313, . . . . . . . . . . 1528, 1613
619, 643, 664, 677, 1315, 1599, 1648, \ccls@InfoFuncEns 1495, 1556
690, 701, 712, 731, 1665, 1666, 1668, \ccls@InfoPFuncEns 1516, 1583
735, 753, 809, 831, 1684, 1686, 1689, \ccls@InfoVFuncEns 1537, 1620
846, 875, 880, 888, 1690, 1692, 1774, \ccls@LINafbg . . . 1901, 1903
900, 919, 942, 997, 1779, 1784, 1812, \ccls@LINDafbg . . 1902, 1904
1014, 1069, 1091, 1323 1813, 1835, 1901, 1902 \ccls@LINDf . 1895, 1899, 1902
\@OUTPUTSOLS . . . . . . 145, 252 \ARCCOS . . . . 810, 1090, 1766 \ccls@LINDg . 1896, 1900, 1902
\@OUTPUTVECTOR . . . . 968, \ARCCOSfunction . . . . . 1765 \ccls@LINf . 1895, 1897, 1901
1074, 1255, 1272, 1456 \ARCCOT . . . . . . . . . 876, 1777 \ccls@LINg . 1896, 1898, 1901
\@POWER . . . . . . 226, 227, 232 \ARCCOTfunction . . . . . 1776 \ccls@lpoly . . . . . . 1636,
\@ROUND . . . . . . . . . . 307, 308 \ARCOSH . . . . . . . . . 889, 1788 1639, 1642, 1645, 1646
\@SCLRDIVVECT . . . . 1347, \ARCOSHfunction . . . . . 1787 \ccls@nminusone . 1885, 1886
1349, 1351, 1356, 1446 \ARCOTH . . . . . . . . . 920, 1799 \ccls@polarx . . . . 1907, 1909

82
\ccls@polary . . . . 1908, 1909 \cctr@@detc . . . . . 1397, 1405 \cctr@outd . . . . . . . . . . .
\ccls@POWDf . . . . . 1883, 1888 \cctr@absval . . . . . . 622, . 1167, 1168, 1173, 1177
\ccls@POWDfn 1886–1888, 1890 623, 667, 668, 680, \cctr@oute . . . . . 1173, 1177
\ccls@POWf . 1883, 1884, 1886 681, 693, 694, 704, 705 \cctr@outf . . . . . 1173, 1177
\ccls@POWfn . . . . . 1884, 1889 \cctr@ae . . 721, 722, 726, 727 \cctr@outg . . . . . 1174, 1178
\ccls@PRODf . . . . . 1830, 1834 \cctr@det . . . 1311–1315, \cctr@outh . . . . . 1174, 1178
\ccls@PRODfg 1834, 1835, 1837 1335, 1336, 1347, \cctr@outi . . . . . 1174, 1178
\ccls@PRODg . . . . . 1831, 1833 1349, 1351, 1372, \cctr@Q . . . . . . 175, 185, 187
\ccls@PROf . 1830, 1832, 1833 1373, 1437, 1438, 1446 \cctr@sign . . . 169–172, 187
\ccls@PROfDg . . . . 1833, 1835 \cctr@deta . . . . . . . . . . . \cctr@size . . . . . . . 953,
\ccls@PROfg . . . . . 1832, 1836 . 1392, 1395, 1399–1401 954, 964, 965, 976,
\ccls@PROg . 1831, 1832, 1834 \cctr@detb . 1393, 1396, 1399 977, 994, 995, 1011,
\ccls@qpoly . . . . . . 1652, \cctr@detc . 1394, 1397, 1400 1012, 1026, 1027,
1655, 1658, 1661, 1662 \cctr@epsilon . . . . . . . 6, 1037, 1038, 1048,
\ccls@qsquaretempg 1848, 1849 396, 612, 716, 741, 1049, 1062, 1063,
\ccls@QUODf . . . . . 1842, 1846 742, 795, 1325, 1150, 1151, 1162,
\ccls@QUODfg . . . . . . . . . . 1340, 1398, 1402, 1163, 1180, 1181,
. 1846, 1847, 1849, 1851 1405, 1410, 1414, 1193, 1194, 1209,
\ccls@QUODg . . . . . 1843, 1845 1418, 1422, 1427, 1439 1210, 1225, 1226,
\ccls@QUOf . 1842, 1844, 1845 \cctr@expminusx . . . . . . . 1237, 1238, 1252,
\ccls@QUOfDg . . . . 1845, 1847 . . . . 674, 675, 687, 688 1253, 1269, 1270,
\ccls@QUOfg . . . . . 1844, 1850 \cctr@expt 631, 632, 639, 640 1285, 1286, 1301,
\ccls@QUOg . . . . . . . . . . . 1302, 1320, 1321,
\cctr@expx 672, 675, 685, 688
. 1843, 1844, 1846, 1848 1374, 1375, 1451, 1452
\cctr@expy 632, 633, 640, 641
\ccls@QUOnum . . . . 1847, 1849 \cctr@sol . . . . . . . . . . . .
\cctr@lengtha . . . . . . . . .
\ccls@SCFaf . . . . . 1866, 1868 . 1138, 1311, 1313, 1315
. . . 4, 157–160, 162,
\ccls@SCFDaf . . . . 1867, 1869 \cctr@solAA . . . . . . . . . . .
163, 194, 200, 201,
\ccls@SCFDf . . . . . 1865, 1867 . . 1093, 1103, 1113,
203, 205, 206, 211, 213
\ccls@SCFf . . . . . 1865, 1866 1134, 1202, 1218,
\ccls@SCVat . . . . . 1874, 1875 \cctr@lengthb . . . . . . . . .
1278, 1294, 1341,
5, 161, 162, 195, 196,
\ccls@SCVDf . 1875, 1876, 1878 1347, 1348, 1360, 1443
200, 204, 205, 212,
\ccls@SCVf . . . . . 1875, 1877 \cctr@solAB . . 1094, 1104,
214, 395, 396, 403,
\ccls@sint . 1594, 1595, 1598 1114, 1202, 1218,
405, 740, 742, 749,
\ccls@SUBDf . . . . . 1820, 1823 1278, 1294, 1341,
751, 793, 795, 806, 808
\ccls@SUBDfg . . . . 1823, 1825 1347, 1348, 1363, 1443
\cctr@log . . . . . . . . 616–618
\ccls@SUBDg . . . . . 1821, 1823 \cctr@solAC . . 1095, 1105,
\ccls@SUBf . . . . . 1820, 1822 \cctr@loga . . . . . . . 733, 735 1115, 1202, 1218,
\ccls@SUBfg . . . . . 1822, 1824 \cctr@logmaxnum . 8, 114, 1278, 1294, 1341,
\ccls@SUBg . . . . . 1821, 1822 623, 668, 681, 694, 705 1347, 1348, 1366, 1443
\ccls@SUMDf . . . . . 1810, 1813 \cctr@logx . . . . . . . 734, 735 \cctr@solBA . . 1096, 1106,
\ccls@SUMDfg . . . . 1813, 1815 \cctr@minust 673, 674, 686, 687 1116, 1203, 1219,
\ccls@SUMDg . . . . . 1811, 1813 \cctr@ndec . . . 176, 177, 186 1279, 1295, 1342,
\ccls@SUMf . . . . . 1810, 1812 \cctr@outa 144, 145, 147, 1349, 1350, 1361, 1444
\ccls@SUMfg . . . . . 1812, 1814 969, 970, 973, 974, \cctr@solBB . . 1097, 1107,
\ccls@SUMg . . . . . 1811, 1812 1167, 1168, 1172, 1176 1117, 1203, 1219,
\ccls@temp . . . . . 1685, 1686 \cctr@outb . . . . 146, 147, 1279, 1295, 1342,
\cctr@@det . . . . . . 1325, 969, 970, 973, 974, 1349, 1350, 1364, 1444
1340, 1373, 1438, 1439 1167, 1168, 1172, 1176 \cctr@solBC . . 1098, 1108,
\cctr@@deta . . . . . 1395, 1398 \cctr@outc . . . . 973, 974, 1118, 1203, 1219,
\cctr@@detb . . . . . 1396, 1402 1167, 1168, 1172, 1176 1279, 1295, 1342,

83
 1349, 1350, 1367, 1444 584, 835, 836, 840, \cctr@Warnbigartanh . . . .
\cctr@solCA . . 1099, 1109, 841, 885–887, 897– . . . . . . . . . 60, 905, 909
1119, 1204, 1220, 899, 912, 914, 929, \cctr@Warncrossprod 131, 1002
1280, 1296, 1343, 931, 1078, 1084, 1089, \cctr@Warndivzero . . 21, 198
1351, 1352, 1362, 1445 1262–1265, 1408, 1414 \cctr@WarnIncLinSys 83, 1379
\cctr@solCB . . 1100, 1110, \cctr@tempC 915, 916, 932, 933 \cctr@WarnIncTDLinSys . .
1120, 1204, 1220, \cctr@tempc . . . . . . . 231, . . . . . . . . . . . 86, 1386
1280, 1296, 1343, 232, 351, 353, 364, \cctr@WarnIndLinSys 90, 1382
1351, 1352, 1365, 1445 366, 451, 452, 564, \cctr@Warninfcotan . . . . .
\cctr@solCC . . 1101, 1111, 565, 913, 915, 930, . . . . . . . . 104, 506, 594
1121, 1204, 1220, 932, 1079, 1088–1090, \cctr@Warninfexp . . 110,
1280, 1296, 1343, 1262–1265, 1409, 1418 624, 669, 682, 695, 706
1351, 1352, 1368, 1445 \cctr@tempD . . . . . . . . . . . \cctr@Warninfexpb . 118, 613
\cctr@tanhden . . . . . . . . . . 167, 174, 179, 181, 184 \cctr@Warninflog . . 125, 717
. . . . 699, 700, 709, 711 \cctr@tempd 168, 174, 182, \cctr@Warninftan . . . . . .
\cctr@tanhnum . . . . . . . . . 184, 242, 251, 1409, 1422 . . 98, 456, 460, 569, 573
. . . . 698, 700, 710, 711 \cctr@tempdif . . . . . 748, 749 \cctr@Warnnoangle . . . . .
\cctr@temp . . . . . . . 255, \cctr@tempexp . . . . . 221, 222 . . . . . . 137, 1082, 1086
258, 264–266, 1068, 1069 \cctr@tempm . . . 340, 344, 345 \cctr@Warnnogcd . . . . 26, 356
\cctr@tempA . . . . . . . 279, 280 \cctr@tempn . . . . . . . 334, \cctr@Warnnointexp . . 36, 223
\cctr@tempa 213, 215, 234, 337, 338, 340, 343–345 \cctr@Warnnoposrad . . 30, 392
236, 238, 291–293, \cctr@tempo . . . . . . . 329, \cctr@Warnround . . . . 15, 311
313–315, 349, 351, 331, 332, 334, 336– \cctr@Warnsingmatrix 73, 1330
352, 354, 362–366, 338, 340, 342, 343, 345 \cctr@WarnsingTDmatrix .
433, 434, 436, 438, \cctr@tempoldw . . . . 743, 748 . . . . . . . . . . . 78, 1344
439, 441, 444, 475, \cctr@tempoldy . . . . 797, 807 \cctr@Warnsmallarcosh 54, 893
476, 478, 482, 483, \cctr@tempp . . . . . . . 324– \cctr@Warnsmallarcoth 66, 925
485, 486, 488, 489, 327, 329–332, 334– \cctr@Warntruncate . . 9, 289
491, 492, 494, 495, 336, 338, 340–342, 345 \cctr@WarnZeroLinSys 94, 1431
646, 647, 649, 652, \cctr@tempq . . . . 174, 175, \COMPOSITIONfunction . 1853
655, 658, 851, 852, 184, 185, 325–327, \CONSTANTfunction . . . 1803
854, 857, 860, 863, 329, 330, 332, 334, \COPY . . . . . 142, 144, 147,
866, 869, 872, 883– 335, 338, 340, 341, 345 150, 153, 156, 169–
885, 895–897, 911– \cctr@tempr . . . . . . . . . . . 172, 175, 176, 181,
913, 928–930, 939, . 174, 178, 179, 181, 184 185, 201, 234, 235,
940, 982, 983, 987– \cctr@tempw . . . . 397–399, 259, 268, 274, 289,
990, 1072–1074, 1077, 402, 403, 739, 743, 301–305, 311, 324–
1080, 1088, 1262– 744, 746–748, 753, 327, 329–332, 334–
1265, 1307, 1308, 796, 798, 800, 802, 804 338, 340–345, 353,
1408, 1410, 1426, 1427 \cctr@tempx 774–776, 782–784 359, 363, 365, 366,
\cctr@tempB 914, 916, 931, 933 \cctr@tempxw . . . . . . 744–746 370, 378, 388, 394,
\cctr@tempb . . . . . . . . . . . \cctr@tempxx 776, 777, 784, 785 397, 401, 411, 413,
. 214, 215, 230, 232, \cctr@tempy 790, 791, 794, 415, 434, 436, 476,
350–353, 355, 359, 797, 800, 801, 804–807 479, 502, 514, 523,
364, 365, 418, 419, \cctr@tempz . . . . . . . 398– 525, 534, 541, 543,
425, 426, 428, 429, 401, 798, 799, 801–803 545, 590, 743, 753,
464, 465, 470, 471, \cctr@Warnbigarccos . . . . 757, 760, 763, 789,
548, 549, 552, 553, . . . . . . . . . 48, 823, 827 794, 796, 797, 803,
556, 557, 559, 560, \cctr@Warnbigarcsin . . . . 813, 816, 819, 849,
577, 580, 581, 583, . . . . . . . . . 42, 767, 771 911, 928, 941, 944,

84
 945, 947, 950, 1093– 874, 917, 934, 1073, \forceqpoly . . . . . . . . . 1659
1101, 1103–1111, 1088, 1089, 1335, \forcevectorfunction . 1622
1138, 1140, 1141, 1357, 1383, 1399, \FRACTIONALPART . . . . . . 274
1143, 1185, 1331, 1400, 1459, 1461– \FRACTIONSIMPLIFY . . . . 376
1332, 1384, 1432, 1464, 1466, 1704,
1649, 1695, 1696, 1716, 1731, 1736, G
1698, 1699, 1701, 1747, 1752, 1755, \GCD . . . . . . . . . 347, 369, 380
1702, 1719, 1757, 1758 1764, 1770, 1775, \GLOBALCOPY . . . . . . . . . . .
\COS . . . . . . 449, 500, 800, 1780, 1786, 1792, . 143, 144–146, 958,
1592, 1721, 1726, 1729 1797, 1802, 1844, 1849 961, 1113–1121, 1155
\COSfunction . . . . 1720, 1907 \DOTPRODUCT . . . . . . . . . . 998
\COSH . . . . . . . . . 665, 699, \Dr . . . . . . . . . . . 1596, 1598 H
710, 1738, 1742, 1745 \Dx . . 1596, 1597, 1601, 1630 \HEAVISIDEfunction . . . 1756
\COSHfunction . . . . . . . 1737 \Dy . . . 1566, 1598, 1599,
\COT . . . . . . . . . . . . . . . . 498 1603, 1632, 1649, I
1664, 1665, 1683, \IDENTITYfunction . . . 1700
\COTAN . . . . . . . . . . . . . 1733
1686, 1688, 1690, \ifcase . . . . . . . . . . 300, 323
\COTANH . . . . . . . . . . . . 1749
1696, 1699, 1702, \INTEGERDIVISION 240, 255, 258
\COTfunction . . . . . . . . 1732
1705, 1706, 1709, \INTEGERPART . . . . . . 221,
\COTH . . . . . . . . . . . . . . . 702
1711–1713, 1716, 259, 273, 279, 291, 313
\COTHfunction . . . . . . . 1748
1719, 1722, 1723, \INTEGERQUOTIENT . . . . . 256
\CROSSPRODUCT . . . . . . . 1015
1726, 1729–1731, \INVERSEMATRIX . . . . . . 1324
\CUBE . . . . . . . . . . . . . . . 218
1734–1736, 1739,
\CUBEfunction . . . . . . . 1710 L
1742, 1745–1747,
1750–1752, 1755, \LCM . . . . . . . . . . . . . . . . 368
D \LENGTHADD . . . . . . . 203, 208
1758, 1763, 1764,
\DEGREESCOS . . . 536, 588, 602 \LENGTHDIVIDE . . . . . . . . 209
1769, 1770, 1775,
\DEGREESCOT . . . . . . . 538, 606 \LENGTHSUBTRACT . . . . . . 207
1780, 1785, 1786,
\DEGREESSIN . . . . . . . . . . . \LINEARCOMBINATIONfunction
1791, 1792, 1797, 1802
. 535, 549, 557, 560, 565 . . . . . . . . . . . . . . 1892
\DEGREESTAN . . . . . . . . . . . E \LOG 616, 713, 727, 887, 899,
. 537, 577, 584, 592, 604 \ensurecpoly . . . . . . . . 1675 914, 915, 931, 932, 1754
\DEGtoRAD . . . . 511, 552, 580 \ensurefunction . . 1498, \LOGfunction . . . . . . . . 1753
\DETERMINANT . 1005–1007, 1553, 1641, 1657, 1676
1305, 1360–1368, \ensurelpoly . . . . . . . . 1640 M
1372, 1392–1394, 1437 \ensurepolarfunction . . . \MATRIXABSVALUE . 1229, 1408
\DIVIDE . . . . . . . . . . 165, . . . . . . . . . . 1519, 1580 \MATRIXADD . . . . . . . . . 1197
215, 230, 372, 381, \ensureqpoly . . . . . . . . 1656 \MATRIXCOPY . 1142, 1168, 1175
382, 398, 400, 439, \ensurevectorfunction . . \MATRIXGLOBALCOPY . . . . .
442, 445, 481, 482, . . . . . . . . . . 1540, 1617 . . . . . 1154, 1167, 1171
484, 485, 487, 488, \EXP . . . . . . . . . . 609, 672, \MATRIXPRODUCT . . . . . . 1289
490, 491, 493, 494, 674, 685, 687, 744, 1718 \MATRIXSIZE . . . . . . 1139,
496, 509, 511, 597, \EXPfunction . . . . . . . . 1717 1150, 1162, 1180,
607, 631, 639, 647, 1193, 1209, 1225,
649, 650, 652, 653, F 1237, 1252, 1285,
655, 656, 658, 659, \FLOOR . . . . . . . . . . . . . . 273 1301, 1320, 1374, 1451
662, 676, 689, 700, \forcecpoly . . . . . . . . . 1678 \MATRIXSUB . . . . . . . . . 1213
711, 721, 735, 745, \forcefunction . . . . . . . . \MATRIXVECTORPRODUCT . . .
775, 783, 801, 835, . 1558, 1644, 1660, 1679 . 1241, 1290, 1291, 1443
840, 854, 857, 860, \forcelpoly . . . . . . . . . 1643 \MAX . . . . . . . . . . . . . 148, 351
863, 866, 869, 872, \forcepolarfunction . . 1585 \MIN . . . . . . . . . . . . . 151, 352

85
\MODULO . . . . . . . . . . 253, 364 \numberFIFTHPI . . . . . . 1463 R
\MULTIPLY . . . . . . . . 156, \numberGOLD . . . . . . . . . 1470 \r . . . . . . . . . . . . 1593, 1595
157, 179, 182, 187, \numberHALFPI . . 411, 413, \RADtoDEG . . . . . . . . . . . 512
191, 217, 218, 231, 417, 421, 451, 455, \RECIPROCALfunction . . 1703
237, 264, 373, 383, 459, 463, 467, 760, \REDUCEDEGREESANGLE . . . 524
441, 444, 447, 512, 763, 779, 787, 813, \REDUCERADIANSANGLE . . . 513
608, 617, 663, 673, 830, 837, 842, 879, 1459 \renewcpoly . . . . . . . . . 1672
686, 726, 778, 786, \numberINVE . . . . . 1466, 1468 \renewfunction . . . 1488,
852, 855, 858, 861, \numberINVETWO . . . . . . 1468 1547, 1638, 1654, 1673
864, 867, 870, 939, \numberINVGOLD . . . . . . 1471 \renewlpoly . . . . . . . . . 1637
981, 982, 986, 987, \numberLOGTEN . . . . . . . 1469 \renewpolarfunction . . . .
989, 1053, 1054, \numberPI . . . . . 425, 464, . . . . . . . . . . 1505, 1574
1057–1059, 1306, 470, 515, 519, 523, \renewqpoly . . . . . . . . . 1653
1307, 1311, 1312, 819, 1457, 1458–1464 \renewvectorfunction . . .
1314, 1333, 1334, \numberQUARTERPI . . . . 1462 . . . . . . . . . . 1526, 1611
1458, 1460, 1593, \ROUND . . . . . . . . . . . . . . 307
\numberSIXTHPI . . . . . . 1464
1595, 1596, 1598,
\numberSQRTFIVE . . . . . 1474
1647, 1663, 1664, S
\numberSQRTTHREE . . . . 1473
1667, 1682, 1683, \SCALARMATRIXPRODUCT . . .
1685, 1687, 1688, \numberSQRTTWO . . . . . . 1472
\numberTHIRDPI . . . . . . 1461 . . . . . . . . . . 1273, 1336
1691, 1706, 1709, \SCALARPRODUCT . . . . . . . .
1712, 1713, 1723, \numberTHREEHALFPI . . . . .
. . . . . . . 415, 424, 1460 . . . . 980, 998, 1068,
1832–1834, 1845, 1079, 1242, 1243,
1846, 1858, 1866, \numberTWOPI . . . . . . . . . .
1246–1248, 1257, 1258
1867, 1874, 1876, 418, 428, 516, 520, 1458
\SCALARVECTORPRODUCT . . .
1887, 1888, 1897–1900 . . . . . . 1052, 1074,
O
1261, 1262, 1264,
N \ONEfunction . . . . . . . . 1694
1274, 1275, 1278–1280
\NeedsTeXFormat . . . . 2, 1479 \or 302–305, 325, 329, 334, 340
\SCALEfunction . . . . . . 1862
\newcpoly . . . . . . . . . . 1669 \SCALEVARIABLEfunction 1871
\newfunction . . . . . . . . . . P \setlength . . . . . . . . . . 206
. . 1494, 1541, 1635, \PackageError 1483, 1490, \SIN . . . . . . 409, 452, 798,
1651, 1670, 1694, 1500, 1508, 1521, 1529 1594, 1722, 1725, 1734
1697, 1700, 1703, \PackageInfo 1496, 1517, 1538 \SINfunction . . . . 1724, 1908
1707, 1710, 1714, \PackageWarning . . . . 10, \SINH . . . . . . . . . 678, 698,
1717, 1720, 1724, 16, 22, 27, 31, 37, 43, 709, 1739, 1741, 1750
1727, 1732, 1737, 49, 55, 61, 67, 74, 79, \SINHfunction . . . . . . . 1740
1740, 1743, 1748, 83, 86, 90, 94, 99, 105, \SOLVELINEARSYSTEM . . . 1379
1753, 1756, 1759, 111, 119, 126, 132, 138 \SQRT . . . . . . . . . 408, 776,
1765, 1771, 1776, \PARAMETRICfunction . . . . 784, 885, 897, 1715,
1781, 1787, 1793, 1798 . . . . . 1909, 1910, 1914 1763, 1769, 1785, 1791
\newlpoly . . . . . . . . . . 1634 \POLARfunction . . . . . . 1906 \SQRTfunction . . . . . . . 1714
\newpolarfunction 1514, 1568 \POWER . . . . . 219, 1884, 1886 \SQUARE . . . . . . . 217, 438,
\newqpoly . . . . . . . . . . 1650 \POWERfunction . . . . . . 1880 633, 641, 646, 851,
\newvectorfunction 1535, 1605 \PRODUCTfunction . . . . . . 883, 895, 1467, 1468,
\numberCOSXLV . . . . . . . 1475 . . . . . 1827, 1907, 1908 1705, 1708, 1711,
\numberCOSXXX . . . . . . . 1476 \ProvidesPackage . . . 3, 1480 1730, 1735, 1746,
\numberE . . . . . . . . . 721, 1751, 1761, 1767,
726, 1465, 1466, 1467 Q 1773, 1778, 1783,
\numberETWO . . . . . . 720, 1467 \QUOTIENTfunction ... 1839 1789, 1795, 1800, 1848

86
\SQUAREfunction . . . . . 1707 1754, 1755, 1757, \VECTORSIZE 943, 953, 964,
\SQUAREROOT . . 385, 408, 1069 1760, 1761, 1766, 976, 994, 1011, 1026,
\SUBTRACT . . . . . . . . 164, 1767, 1772, 1773, 1037, 1048, 1062, 1269
192, 249, 267, 280, 1777, 1778, 1782, \VECTORSUB . . . . . . 1030,
402, 425, 428, 440, 1783, 1788, 1789, 1214, 1215, 1218–1220
443, 446, 470, 483, 1794, 1795, 1799, 1800
486, 489, 492, 495, \TAN . . . . . . . . 453, 504, 1728 X
520, 531, 556, 559, \TANfunction . . . . . . . . 1727 \x . . . 1593, 1599, 1600, 1629
583, 661, 688, 728, \TANH . . . . . . . . . . . 691, 1744
739, 747, 748, 774, \TANHfunction . . . . . . . 1743 Y
779, 787, 799, 802, \textit . . . . . . . . . . . . . 737 \y . . . . 1565, 1595, 1597,
804, 830, 837, 842, \TRANSPOSEMATRIX . . . . 1184 1602, 1631, 1647,
879, 896, 913, 916, \TRUNCATE . . . . . . . . . . . 285 1648, 1663, 1664,
930, 933, 1308, \TWOVECTORSANGLE . . . . 1075 1666–1668, 1682–
1597, 1762, 1768, 1684, 1687, 1689,
1790, 1796, 1801, U 1691, 1692, 1695,
1822, 1823, 1847, 1885 \UNITVECTOR . . . . . . . . . 1070 1698, 1701, 1704,
\SUBTRACTfunction . . . 1817 1705, 1708, 1712,
\SUMfunction . . . . . . . . 1807 V 1715, 1716, 1718,
\VECTORABSVALUE . . . . . . . 1719, 1721, 1725,
T . . . . . 1041, 1230, 1233 1728, 1733, 1738,
\t . . . . 1563, 1590, 1592, \VECTORADD . . 1016, 1031, 1741, 1744, 1749,
1594, 1627, 1647, 1034, 1198, 1199, 1754, 1757, 1760,
1663, 1667, 1682, 1202–1204, 1263, 1265 1766, 1772, 1777,
1687, 1688, 1691, \VECTORCOPY . . . 946, 970, 974 1782, 1788, 1794, 1799
1701, 1704, 1708, \VECTORfunction . . . . . 1914 \yy . . . 1761–1763, 1767–
1709, 1711, 1712, \VECTORGLOBALCOPY . . . . . 1769, 1773–1775,
1715, 1718, 1721, . . . . . . . . 957, 969, 973 1778–1780, 1783–
1722, 1725, 1726, \VECTORMATRIXPRODUCT . . . 1785, 1789–1791,
1728, 1729, 1733, . . . . . 1256, 1294–1296 1795–1797, 1800–1802
1734, 1738, 1739, \VECTORNORM . . . . . . 1066,
1741, 1742, 1744, 1072, 1077, 1078, 1426 Z
1745, 1749, 1750, \VECTORPRODUCT . . . 999, 1015 \ZEROfunction . . . . . . . 1697

87