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<H1><A NAME="SEC67" HREF="maxima_toc.html#TOC67">Numerical</A></H1>
<H2><A NAME="SEC68" HREF="maxima_toc.html#TOC68">Introduction to Numerical</A></H2>
<H2><A NAME="SEC69" HREF="maxima_toc.html#TOC69">DCADRE</A></H2>
<P>
The following is obsolete.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
<PRE>
LOADFILE("imsl")$
</PRE>
<P>
For a demo of this package, do
<PRE>
batch("dcadre.mc");
</PRE>
<P>
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 --
</P>
<P>
ROMBERG_AERR [Default 1.0E-5] -- Desired absolute error in answer.
</P>
<P>
ROMBERG_RERR [Default 0.0] -- Desired relative error in the answer.
</P>
<P>
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.)
</P>
<P>
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.
</P>
<P>
Purpose (modified version of the IMSL documentation)
----------------------------------------------------
</P>
<P>
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
</P>
<P>
DCADRE such that:
</P>
<PRE>
| / B | [ | / B | ]
| [ | [ | [ | ]
| I F(x)dx - DCADRE | <= max [ ROMBERG_AERR, ROMBERG_RERR * | I F(x)dx | ]
| ] | [ | ] | ]
| / A | [ | / A | ]
</PRE>
<P>
Algorithm (modified version of the IMSL documentation)
</P>
<P>
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)
</P>
<UL>
<LI>
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.
<LI>
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)).
<LI>
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.
<LI>
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.
</UL>
<P>
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.
</P>
<H2><A NAME="SEC70" HREF="maxima_toc.html#TOC70">ELLIPT</A></H2>
<P>
- 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
</P>
<PRE>
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
</PRE>
<H2><A NAME="SEC71" HREF="maxima_toc.html#TOC71">FOURIER</A></H2>
<P>
- 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.
</P>
<H2><A NAME="SEC72" HREF="maxima_toc.html#TOC72">NDIFFQ</A></H2>
<P>
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:
</P>
<PRE>
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[0]:5.0; /* Initial condition */
Runge_Kutta(F,X,E); /* Solve it */
Graph2(X,E); /* Graph the solution */
</PRE>
<P>
p.s. Runge_Kutta(F,X,E,E_Prime) would be the call for a second-order
equation.
</P>
<H2><A NAME="SEC73" HREF="maxima_toc.html#TOC73">Definitions for Numerical</A></H2>
<P>
<DL>
<DT><U>Function:</U> <B>FFT</B> <I>(real-array, imag-array)</I>
<DD><A NAME="IDX550"></A>
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:
<PRE>
ARRAY(<ary>,FLOAT,<dim1>); or
ARRAY(<ary>,FLOAT,<dim1>,<dim2>);
</PRE>
<P>
For 1D arrays
<PRE>
<dim1> = 2^n-1
</PRE>
<P>
and for 2D arrays
<PRE>
<dim1>=<dim2>=2^n-1
</PRE>
<P>
(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); .
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>FORTINDENT</B>
<DD><A NAME="IDX551"></A>
default: [0] - 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.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>FORTMX</B> <I>(name,matrix)</I>
<DD><A NAME="IDX552"></A>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>FORTRAN</B> <I>(exp)</I>
<DD><A NAME="IDX553"></A>
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[0] 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.
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>FORTSPACES</B>
<DD><A NAME="IDX554"></A>
default: [FALSE] - if TRUE, the FORTRAN command fills out
to 80 columns using spaces.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>HORNER</B> <I>(exp, var)</I>
<DD><A NAME="IDX555"></A>
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);)
<PRE>
(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
</PRE>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>IFT</B> <I>(real-array, imag-array)</I>
<DD><A NAME="IDX556"></A>
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); .
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>INTERPOLATE</B> <I>(func,x,a,b)</I>
<DD><A NAME="IDX557"></A>
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
</P>
<PRE>
INTERPOLATE(SIN(X)=X/2,X,%PI,.1);
is equivalent to
INTERPOLATE(SIN(X)=X/2,X,.1,%PI);
</PRE>
<P>
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.
</P>
<PRE>
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
</PRE>
<P>
There is also a Newton method interpolation routine, do DESCRIBE(NEWTON); .
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>INTPOLABS</B>
<DD><A NAME="IDX558"></A>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>INTPOLERROR</B>
<DD><A NAME="IDX559"></A>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Variable:</U> <B>INTPOLREL</B>
<DD><A NAME="IDX560"></A>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>NEWTON</B> <I>(exp,var,X0,eps)</I>
<DD><A NAME="IDX561"></A>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>POLARTORECT</B> <I>(magnitude-array, phase-array)</I>
<DD><A NAME="IDX562"></A>
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
</P>
<PRE>
<real>=<magnitude>*COS(<phase>) ==>
<imaginary>=<magnitude>*SIN(<phase>
</PRE>
<P>
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.
</P>
</DL>
<P>
<DL>
<DT><U>Function:</U> <B>RECTTOPOLAR</B> <I>(real-array, imag-array)</I>
<DD><A NAME="IDX563"></A>
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.
</P>
</DL>
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