# Numerical

## Introduction to Numerical

The following is obsolete. To make an interface to fortran libraries in the current MAXIMA look at the examples in "maxima/src/fortdef.lsp" - The IMSL version of Romberg integration is now available in Macsyma. For documentation, Do PRINTFILE(DCADRE,USAGE,IMSL1); . For a demo, do batch("dcadre.mc"); This is a numerical integration package using cautious, adaptive Romberg extrapolation. The DCADRE package is written to call the IMSL fortran library routine DCADRE. This is documentation for that program. Send bugs/comments to KMP To load this package, do

```  LOADFILE("imsl")\$
```

For a demo of this package, do

```  batch("dcadre.mc");
```

The worker function takes the following syntax: IMSL_ROMBERG(fn,low,hi) where fn is a function of 1 argument; low and hi should be the lower and upper bounds of integration. fn must return floating point values. IMSL_ROMBERG(exp,var,low,hi) where exp should be integrated over the range var=low to hi. The result of evaluating exp must always be a floating point number. FAST_IMSL_ROMBERG(fn,low,hi) This function does no error checking but may achieve a speed gain over the IMSL_ROMBERG function. It expects that fn is a Lisp function (or translated Macsyma function) which accepts a floating point argument and that it always returns a floating point value. Returns either [SUCCESS, answer, error] where answer is the result of the integration and error is the estimated bound on the absolute error of the output, DCADRE, as described in PURPOSE below. or [WARNING, n, answer, error] where n is a warning code, answer is the answer, and error is the estimated bound on the absolute error of the output, DCADRE, as described in PURPOSE below. The following warnings may occur: 65 = One or more singularities were successfully handled. 66 = In some subinterval(s), the estimate of the integral was accepted merely because the estimated error was small, even though no regular behavior was recognized. or [ERROR, errorcode] where error code is the IMSL-generated error code. The following error codes may occur: 131 = Failure due to insufficient internal working storage. 132 = Failure. This may be due to too much noise in function (relative to the given error requirements) or due to an ill-behaved integrand. 133 = RERR is greater than 0.1 or less than 0.0 or is too small for the precision of the machine. The following flags have an influence upon the operation of IMSL_ROMBERG --

ROMBERG_AERR [Default 1.0E-5] -- Desired absolute error in answer.

ROMBERG_RERR [Default 0.0] -- Desired relative error in the answer.

Note: If IMSL signals an error, a message will be printed on the user's console stating the nature of the error. (This error message may be supressed by setting IMSLVERBOSE to FALSE.)

Note: Because this uses a translated Fortran routine, it may not be recursively invoked. It does not call itself, but the user should be aware that he may not type ^A in the middle of an IMSL_ROMBERG computation, begin another calculation using the same package, and expect to win -- IMSL_ROMBERG will complain if it was already doing one project when you invoke it. This should cause minimal problems.

Purpose (modified version of the IMSL documentation) ----------------------------------------------------

DCADRE attempts to solve the following problem: Given a real-valued function F of one argument, two real numbers A and B, find a number

```|   / B               |        [                              | / B      | ]
|   [                 |        [                              | [        | ]
|   I F(x)dx - DCADRE | <= max [ ROMBERG_AERR, ROMBERG_RERR * | I F(x)dx | ]
|   ]                 |        [                              | ]        | ]
|   / A               |        [                              | / A      | ]
```

Algorithm (modified version of the IMSL documentation)

This routine uses a scheme whereby DCADRE is computed as the sum of estimates for the integral of F(x) over suitably chosen subintervals of the given interval of integration. Starting with the interval of integration itself as the first such subinterval, cautious Romberg extrapolation is used to find an acceptable estimate on a given subinterval. If this attempt fails, the subinterval is divided into two subintervals of equal length, each of which is considered separately. Programming Notes (modified version of the IMSL documentation)

• 1. DCADRE (the translated-Fortran base for IMSL_ROMBERG) can, in many cases, handle jump discontinuities and certain algebraic discontinuities. See reference for full details.
• 2. The relative error parameter ROMBERG_RERR must be in the interval [0.0,0.1]. For example, ROMBERG_RERR=0.1 indicates that the estimate of the intergral is to be correct to one digit, where as ROMBERG_RERR=1.0E-4 calls for four digits of accuracy. If DCADRE determines that the relative accuracy requirement cannot be satisfied, IER is set to 133 (ROMBERG_RERR should be large enough that, when added to 100.0, the result is a number greater than 100.0 (this will not be true of very tiny floating point numbers due to the nature of machine arithmetic)).
• 3. The absolute error parameter, ROMBERG_AERR, should be nonnegative. In order to give a reasonable value for ROMBERG_AERR, the user must know the approximate magnitude of the integral being computed. In many cases, it is satisfactory to use AERR=0.0. In this case, only the relative error requirement is satisfied in the compuatation.
• 4. We quote from the reference, "A very cautious man would accept DCADRE only if IER [the warning or error code] is 0 or 65. The merely reasonable man would keep the faith even if IER is 66. The adventurous man is quite often right in accepting DCADRE even if the IER is 131 or 132." Even when IER is not 0, DCADRE returns the best estimate that has been computed.

For references on this technique, see de Boor, Calr, "CADRE: An Algorithm for Numerical Quadrature," Mathematical Software (John R. Rice, Ed.), New York, Academic Press, 1971, Chapter 7.

## ELLIPT

- A package on the SHARE directory for Numerical routines for Elliptic Functions and Complete Elliptic Integrals. (Notation of Abramowitz and Stegun, Chs 16 and 17) Do LOAD(ELLIPT); to use this package. At present all arguments MUST be floating point. You'll get nonsense otherwise. Be warned. The functions available are: Jacobian elliptic functions

```AM(U,M) - amplitude with modulus M
AM1(U,M1) - amplitude with complementary modulus M1
AM(U,M):=AM1(U,1-M); so use AM1 if M ~ 1
SN(U,M):=SIN(AM(U,M));
CN(U,M):=COS(AM(U,M));
DN(U,M):=SQRT(1-M*SN(U,M)^2);
(These functions come defined like this.  Others CD, NS etc.  may be
similarly defined.)
Complete Elliptic Integrals
ELLIPTK(M) - Complete elliptic integral of first kind
ELLIPTK1(M1) - Same but with complementary modulus.
ELLIPTK(M):=ELLIPTK1(1-M); so use if M ~ 1
ELLIPTE(M) - Complete elliptic integral of second kind
ELLIPTE1(M1) - Same but with complementary modulus.
ELLIPTE(M):=ELLIPTE1(1-M); so use if M ~ 1
```

## FOURIER

- There is a Fast Fourier Transform package, do DESCRIBE(FFT) for details. There is also a Fourier Series package. It may be loaded with LOAD(FOURIE). It will also calculate Fourier integral coefficients and has various other functions to do such things as replace all occurrences of F(ARG) by ARG in expression (like changing ABS(a*x+b) to a*x+b). Do PRINTFILE(FOURIE,USAGE,DSK,SHARE1); for a list of the functions included.

## NDIFFQ

a package residing on the SHARE directory for numerical solutions of differential equations. LOAD("NDIFFQ"); will load it in for use. An example of its use would be:

```Define_Variable(N,0.3,FLOAT);
Define_Variable(H,0.175,FLOAT);
F(X,E):=(Mode_Declare([X,E],FLOAT),N*EXP(X)/(E+X^(2*H)*EXP(H*X)));
Compile(F);
Array([X,E],FLOAT,35);
Init_Float_Array(X,1.0E-3,6.85); /* Fills X with the interval */
E:5.0;                        /* Initial condition */
Runge_Kutta(F,X,E);              /* Solve it */
Graph2(X,E);                     /* Graph the solution */
```

p.s. Runge_Kutta(F,X,E,E_Prime) would be the call for a second-order equation.

## Definitions for Numerical

Function: FFT (real-array, imag-array)
Fast Fourier Transform. This package may be loaded by doing LOAD(FFT); There is also an IFT command, for Inverse Fourier Transform. These functions perform a (complex) fast fourier transform on either 1 or 2 dimensional FLOATING-POINT arrays, obtained by:
```ARRAY(<ary>,FLOAT,<dim1>); or
ARRAY(<ary>,FLOAT,<dim1>,<dim2>);
```

For 1D arrays

```<dim1> = 2^n-1
```

and for 2D arrays

```<dim1>=<dim2>=2^n-1
```

(i.e. the array is square). (Recall that MACSYMA arrays are indexed from a 0 origin so that there will be 2^n and (2^n)^2 arrays elements in the above two cases.) This package also contains two other functions, POLARTORECT and RECTTOPOLAR. Do DESCRIBE(cmd) for details. For details on the implementation, do PRINTFILE(FFT,USAGE,SHARE); .

Variable: FORTINDENT
default:  - controls the left margin indentation of expressions printed out by the FORTRAN command. 0 gives normal printout (i.e. 6 spaces), and positive values will causes the expressions to be printed farther to the right.

Function: FORTMX (name,matrix)
converts a MACSYMA matrix into a sequence of FORTRAN assignment statements of the form name(i,j)=<corresponding matrix element>. This command is now obsolete. FORTMX(name,matrix); may now be done as FORTRAN(name=matrix);. (If "name" is bound, FORTRAN('name=matrix); may be necessary.) Please convert code that uses the FORTMX command as it may be flushed some day.

Function: FORTRAN (exp)
converts exp into a FORTRAN linear expression in legal FORTRAN with 6 spaces inserted at the beginning of each line, continuation lines, and ** rather than ^ for exponentiation. When the option FORTSPACES[FALSE] is TRUE, the FORTRAN command fills out to 80 columns using spaces. If FORTRAN is called on a bound symbolic atom, e.g. FORTRAN(X); where X:A*B\$ has been done, then X={value of X}, e.g. X=A*B will be generated. In particular, if e.g. M:MATRIX(...); has been done, then FORTRAN(M); will generate the appropriate assignment statements of the form name(i,j)=<corresponding matrix element>. FORTINDENT controls the left margin of expressions printed out, 0 is the normal margin (i.e. indented 6 spaces), increasing it will cause the expression to be printed further to the right.

Variable: FORTSPACES
default: [FALSE] - if TRUE, the FORTRAN command fills out to 80 columns using spaces.

Function: HORNER (exp, var)
will convert exp into a rearranged representation as in Horner's rule, using var as the main variable if it is specified. Var may also be omitted in which case the main variable of the CRE form of exp is used. HORNER sometimes improves stability if expr is to be numerically evaluated. It is also useful if MACSYMA is used to generate programs to be run in FORTRAN (see DESCRIBE(STRINGOUT);)
```(C1) 1.0E-20*X^2-5.5*X+5.2E20;
2
(D1)                   1.0E-20 X  - 5.5 X + 5.2E+20
(C2) HORNER(%,X),KEEPFLOAT:TRUE;
(D2)                  X (1.0E-20 X - 5.5) + 5.2E+20
(C3) D1,X=1.0E20;
ARITHMETIC OVERFLOW
(C4) D2,X=1.0E20;
(D4)                          6.9999999E+19

```

Function: IFT (real-array, imag-array)
Inverse Fourier Transform. Do LOAD(FFT); to load in this package. These functions (FFT and IFT) perform a (complex) fast fourier transform on either 1 or 2 dimensional FLOATING-POINT arrays, obtained by: ARRAY(<ary>,FLOAT,<dim1>); or ARRAY(<ary>,FLOAT,<dim1>,<dim2>); For 1D arrays <dim1> must equal 2^n-1, and for 2D arrays <dim1>=<dim2>=2^n-1 (i.e. the array is square). (Recall that MACSYMA arrays are indexed from a 0 origin so that there will be 2^n and (2^n)^2 arrays elements in the above two cases.) For details on the implementation, do PRINTFILE(FFT,USAGE,SHARE); .

Function: INTERPOLATE (func,x,a,b)
finds the zero of func as x varies. The last two args give the range to look in. The function must have a different sign at each endpoint. If this condition is not met, the action of the of the function is governed by INTPOLERROR[TRUE]). If INTPOLERROR is TRUE then an error occurs, otherwise the value of INTPOLERROR is returned (thus for plotting INTPOLERROR might be set to 0.0). Otherwise (given that MACSYMA can evaluate the first argument in the specified range, and that it is continuous) INTERPOLATE is guaranteed to come up with the zero (or one of them if there is more than one zero). The accuracy of INTERPOLATE is governed by INTPOLABS[0.0] and INTPOLREL[0.0] which must be non-negative floating point numbers. INTERPOLATE will stop when the first arg evaluates to something less than or equal to INTPOLABS or if successive approximants to the root differ by no more than INTPOLREL * <one of the approximants>. The default values of INTPOLABS and INTPOLREL are 0.0 so INTERPOLATE gets as good an answer as is possible with the single precision arithmetic we have. The first arg may be an equation. The order of the last two args is irrelevant. Thus

```INTERPOLATE(SIN(X)=X/2,X,%PI,.1);
is equivalent to
INTERPOLATE(SIN(X)=X/2,X,.1,%PI);
```

The method used is a binary search in the range specified by the last two args. When it thinks the function is close enough to being linear, it starts using linear interpolation. An alternative syntax has been added to interpolate, this replaces the first two arguments by a function name. The function MUST be TRANSLATEd or compiled function of one argument. No checking of the result is done, so make sure the function returns a floating point number.

```F(X):=(MODE_DECLARE(X,FLOAT),SIN(X)-X/2.0);
INTERPOLATE(SIN(X)-X/2,X,0.1,%PI)       time= 60 msec
INTERPOLATE(F(X),X,0.1,%PI);            time= 68 msec
TRANSLATE(F);
INTERPOLATE(F(X),X,0.1,%PI);            time= 26 msec
INTERPOLATE(F,0.1,%PI);                 time=  5 msec
```

There is also a Newton method interpolation routine, do DESCRIBE(NEWTON); .

Variable: INTPOLABS
default: [0.0] - The accuracy of the INTERPOLATE command is governed by INTPOLABS[0.0] and INTPOLREL[0.0] which must be non-negative floating point numbers. INTERPOLATE will stop when the first arg evaluates to something less than or equal to INTPOLABS or if successive approximants to the root differ by no more than INTPOLREL * <one of the approximants>. The default values of INTPOLABS and INTPOLREL are 0.0 so INTERPOLATE gets as good an answer as is possible with the single precision arithmetic we have.

Variable: INTPOLERROR
default: [TRUE] - Governs the behavior of INTERPOLATE. When INTERPOLATE is called, it determines whether or not the function to be interpolated satisfies the condition that the values of the function at the endpoints of the interpolation interval are opposite in sign. If they are of opposite sign, the interpolation proceeds. If they are of like sign, and INTPOLERROR is TRUE, then an error is signaled. If they are of like sign and INTPOLERROR is not TRUE, the value of INTPOLERROR is returned. Thus for plotting, INTPOLERROR might be set to 0.0.

Variable: INTPOLREL
default: [0.0] - The accuracy of the INTERPOLATE command is governed by INTPOLABS[0.0] and INTPOLREL[0.0] which must be non-negative floating point numbers. INTERPOLATE will stop when the first arg evaluates to something less than or equal to INTPOLABS or if successive approximants to the root differ by no more than INTPOLREL * <one of the approximants>. The default values of INTPOLABS and INTPOLREL are 0.0 so INTERPOLATE gets as good an answer as is possible with the single precision arithmetic we have.

Function: NEWTON (exp,var,X0,eps)
The file NEWTON 1 on the SHARE directory contains a function which will do interpolation using Newton's method. It may be accessed by LOAD(NEWTON); . The Newton method can do things that INTERPOLATE will refuse to handle, since INTERPOLATE requires that everything evaluate to a flonum. Thus NEWTON(x^2-a^2,x,a/2,a^2/100); will say that it can't tell if flonum*a^2<a^2/100. Doing ASSUME(a>0); and then doing NEWTON again works. You get x=a+<small flonum>*a which is symbolic all the way. INTERPOLATE(x^2-a^2,x,a/2,2*a); complains that .5*a is not flonum... An adaptive integrator which uses the Newton-Cotes 8 panel quadrature rule is available in SHARE1;QQ FASL. Do DESCRIBE(QQ) for details.

Function: POLARTORECT (magnitude-array, phase-array)
converts from magnitude and phase form into real and imaginary form putting the real part in the magnitude array and the imaginary part into the phase array

```<real>=<magnitude>*COS(<phase>) ==>
<imaginary>=<magnitude>*SIN(<phase>
```

This function is part of the FFT package. Do LOAD(FFT); to use it. Like FFT and IFT this function accepts 1 or 2 dimensional arrays. However, the array dimensions need not be a power of 2, nor need the 2D arrays be square.

Function: RECTTOPOLAR (real-array, imag-array)
undoes POLARTORECT. The phase is given in the range from -%PI to %PI. This function is part of the FFT package. Do LOAD(FFT); to use it. Like FFT and IFT this function accepts 1 or 2 dimensional arrays. However, the array dimensions need not be a power of 2, nor need the 2D arrays be square.