The REDUCE Root Finding Package
Mod 1.91, 16 May 1990
Stanley L. Kameny
valley!stan@rand.org
Introduction.
The Root Finding package is designed so that it can be used as an
independent package, or it can be integrated with and called by Solve.
This document describes the package in its independent use. It can be
used to find some or all of the roots of polynomials with real or
complex coefficients, to the accuracy specified by the user.
------------------------------------------------------------------------
Modules.
The file roots.red. contains the following modules needed for use with
REDUCE 3.4:
ROOTS - header module
BFDOER \
BFDOER2 | - these modules can be loaded in any order, but
COMPLXP | - after the math file.
ALLROOT |
REALROOT /
The math module available with REDUCE 3.4, is also required, or else
the roots file cannot be loaded or operated.
------------------------------------------------------------------------
Top Level Functions
The top level functions can be called either as symbolic operators from
algebraic mode, or they can be called directly from symbolic mode with
symbolic mode arguments. Outputs are expressed in forms that print out
correctly in algebraic mode.
Functions which refer to real roots only:
Three top level functions refer only to real roots. Each of these
functions can receive 1, 2 or 3 arguments.
The first argument is the polynomial p, which can be complex and can
have multiple or zero roots. If arg2 and arg3 are not present, all real
roots are found. If the additional arguments are present, they restrict
the region of consideration.
If arguments are (p,arg2) then
(Arg2 must be POSITIVE or NEGATIVE) If arg2=NEGATIVE then only
negative roots of p are included; if arg2=POSITIVE then only positive
roots of p are included. Zero roots are excluded.
If arguments are (p,arg2,arg3)
(Arg2 and Arg3 must be r (a real number) or EXCLUDE r or a member of
the list {POSITIVE,NEGATIVE,INFINITY,-INFINITY}. EXCLUDE r causes the
value r to be excluded from the region. The order of the sequence
arg2, arg3 is unimportant. Assuming that arg2 <= arg3 if both are
numeric, then
{-INFINITY,INFINITY} is equivalent to {} and represents all roots;
{arg2,NEGATIVE} represents -infinity < r < arg2;
{arg2,POSITIVE) represents arg2 < r < infinity;
In each of the following, replacing an arg with EXCLUDE arg converts
the corresponding inclusive <= to the exclusive <
{arg2,-INFINITY} represents -infinity < r <= arg2;
{arg2,INFINITY} represents arg2 <= r < infinity;
{arg2,arg3} represents arg2 <= r <= arg3;
If zero is in the interval, zero root is included.
REALROOTS --- This function finds the real roots of the polynomial p,
using the REALROOT package to isolate real roots by the method of Sturm
sequences, then polishing the root to the desired accuracy. Precision
of computation is guaranteed to be sufficient to separate all real roots
in the specified region. (cf. MULTIROOT for treatment of multiple
roots.)
ISOLATER --- This function produces a list of rational intervals, each
containing a single real root of the polynomial p, within the specified
region, but does not find the roots.
RLROOTNO --- This function computes the number of real roots of p in
the specified region, but does not find the roots.
Functions which return both real and complex roots:
ROOTS p; This is the main top level function of the roots package. It
will find all roots, real and complex, of the the polynomial p to an
accuracy sufficient to separate them. The value returned by ROOTS is a
list of equations for all roots. In addition, ROOTS stores separate
lists of real roots and complex roots in the global variables ROOTSREAL
and ROOTSCOMPLEX.
NEARESTROOT(p,s); This top level function uses an iterative method to
find the root to which the method converges given the initial starting
origin s, which can be complex. If there are several roots in the
vicinity of s and s is not significantly closer to one root than it is
to all others, the convergence could arrive at a root which is not truly
the nearest root. This function should therefore be used only when the
user is certain that there is only one root in the immediate vicinity of
the starting point s.
FIRSTROOT p; Equivalent to NEARESTROOT(p,0).
Other top level function:
CSIZE p; This function will determine the maximum coefficient size of
the polynomial p. The initial precision used in root finding is at
least 2+CSIZE p (in some cases significantly greater, as determined by
the heuristic function CALCPREC.)
GETROOT(n,rr); If rr has the form of the output of ROOTS, REALROOTS, or
NEARESTROOTS; GETROOT returns the rational, real, or complex value of
the root equation. Error occurs if n<1 or n>the number of roots in rr.
MKPOLY rr; This function can be used to reconstruct a polynomial
whose root equation list is rr and whose denominator is 1. Thus one can
verify that
if rr := ROOTS p; then rr1 := ROOTS MKPOLY rr; should result in
rr1 = rr
(This will be true if MULRITOOT and RATROOT are ON, and BIGFLOAT and
FLOAT are off.)
However, MKPOLY rr - NUM p = 0 will be true iff all roots of p
have been computed exactly.
Functions available for diagnostic or instructional use only:
GFNEWT(p,r,cpx); This function will do a single pass through the
function GFNEWTON for polynomial p and root r. If cpx=T, then any
complex part of the root will be kept, no matter how small.
GFROOT(p,r,cpx); This function will do a single pass through the
function GFROOTFIND for polynomial p and root r. If cpx=T, then any
complex part of the root will be kept, no matter how small.
ROOTS2 p; The same as ROOTS p, except that if an abort occurs, the
roots already found will be printed and then ROOTS2 will be applied to
the deflated polynomial which exists at that point. (Note: there is no
known polynomial on which ROOTS aborts.)
------------------------------------------------------------------------
Switches Used in Input.
The input of polynomials in algebraic mode is sensitive to the switches
COMPLEX, FLOAT and BIGFLOAT. The correct choice of input method is
important since incorrect choices will result in undesirable truncation
or rounding of the input coefficients.
Truncation or rounding will occur if FLOAT or BIGFLOAT is on and one of
the following is true:
a) a coefficient is entered in floating point form or rational form.
b) COMPLEX is on and a coefficient is imaginary or complex.
Therefore, to avoid undesirable truncation or rounding, then:
c) both FLOAT and BIGFLOAT should be off and input should be in
integer or rational form; or
d) FLOAT can be on if it is acceptable to truncate or round input to
the machine-dependent precision limit, which may be quite small; or
e) BIGFLOAT can be on if PRECISION is set to a value large enough to
prevent undesired rounding.
integer and complex modes: (off float, bigfloat) any real polynomial
can be input using integer coefficients of any size; integer or
rational coefficients can be used to input any real or complex
polynomial, independent of the setting of the switch COMPLEX. These are
the most versatile input modes, since any real or complex polynomial can
be input exactly.
modes float and complex-float: (on float) polynomials can be input using
integer coefficients of any size. Floating point coefficients will be
truncated or rounded, to a size dependent upon the system. If complex
is on, real coefficients can be input to any precision using integer
form, but coefficients of imaginary parts of complex coefficients will
be rounded or truncated.
modes bigfloat and big-complex: (on bigfloat) the setting of precision
determines the precision of all coefficients except for real
coefficients input in integer form. Floating point coefficients will be
truncated by the system to a size dependent upon the system, the same as
floating point coefficients in float mode. If precision is set high
enough, any real or complex polynomial can be input exactly provided
that coefficients are input in integer or rational form.
Internal and Output Use of Switches.
REDUCE arithmetic mode switches BIGFLOAT, FLOAT, and COMPLEX. These
switches are returned in the same state in which they were set
initially, (barring catastrophic error.)
COMPLEX --The Root Finding Package controls the switch COMPLEX
internally, turning the switch on if it is processing a complex
polynomial. (However, if COMPLEX is on, algebraic mode input may not
work correctly in modes COMPLEX_FLOAT or BIG_COMPLEX, so it is best to
use integer or rational input only. See example 62 of roots.tst for a
way to get this to work.) For a polynomial with real coefficients, the
starting point argument for NEARESTROOT can be given in algebraic mode
in complex form as rl + im * I and will be handled correctly,
independent of the setting of the switch COMPLEX. Complex roots will be
computed and printed correctly regardless of the setting of the switch
COMPLEX. However, if COMPLEX is off, the imaginary part will print out
ahead of the real part, while the reverse order will be obtained if
COMPLEX is on.
FLOAT, BIGFLOAT --If the switch AUTOMODE (Default ON) is ON, the Root
Finding package performs computations using the arithmetic mode that is
required at the time, which may be integer, Gaussian integer, float,
bigfloat, complex float or complex bigfloat. Switch BFTAG is used
internally to govern the mode of computation and :prec: is adjusted
whenever necessary. The initial position of switches FLOAT and BIGFLOAT
are ignored. At output, these switches will emerge in their initial
positions. Outputs will be printed out in float format only if the
float format of the Lisp system will properly print out quantities of
the required accuracy. Otherwise, the printout will be in bigfloat
format. (See also the paragraph describing AUTOMODE.)
Root Package Switches RATROOT, MULTIROOT, TRROOT, ROOTMSG,
(plus diagnostic switch AUTOMODE.)
(switches ISOROOT and ACCROOT, present in earlier versions, have been
eliminated.)
RATROOT --(Default OFF) If RATROOT is on all root equations are output
in rational form. Assuming that the mode is COMPLEX (ie. FLOAT and
BIGFLOAT are both off,) the root equations are guaranteed to be able to
be input into REDUCE without truncation or rounding errors. (Cf. the
function MKPOLY described above.)
MULTIROOT --(Default ON) Whenever the polynomial has complex
coefficients or has real coefficients and has multiple roots, as
determined by the Sturm function, the function SQFRF is called
automatically to factor the polynomial into square-free factors. If
MULTIROOT is on, the multiplicity of the roots will be indicated in the
output of ROOTS or REALROOTS by printing the root output repeatedly,
according to its multiplicity. If MULTIROOT is off, each root will be
printed once, and all roots should be normally be distinct. (Two
identical roots should not appear. If the initial precision of the
computation or the accuracy of the output was insufficient to separate
two closely-spaced roots, the program attempts to increase accuracy
and/or precision if it detects equal roots. If however, if the initial
accuracy specified was too low, and it was possible to separate the
roots, the program will abort.)
TRROOT --(Default OFF) If switch TRROOT is on, trace messages are
printed out during the course of root determination, to show the
progress of solution.
ROOTMSG --(Default OFF) If switch ROOTMSG is on in addition to switch
TRROOT, additional messages are printed out to aid in following the
progress of Laguerre and Newton complex iteration. These messages are
intended for debugging use primarily.
NOTE: the switch AUTOMODE is included mainly for diagnostic purposes.
If it is changed from its default setting, the automatic determination
of computation modes is bypassed, and correct root determination may not
be achieved!
AUTOMODE --(Default ON) If switch AUTOMODE is on, then, independent of
the user setting of the switch BIGFLOAT, all floating point computations
are carried out in floating point mode (rather than bigfloat) if the
system floating point mode has sufficient precision at that point in the
computation. If AUTOMODE is off and the user setting of BIGFLOAT is on,
bigfloat computations are used for all floating point computations. The
default setting of AUTOMODE is ON, in order to speed up computations and
guarantee that the exact input polynomial is evaluated.
------------------------------------------------------------------------
Operational Parameters and Parameter Setting.
ROOTACC# --(Default 6) This parameter can be set using the function
ROOTACC n; which causes ROOTACC# to be set to MAX(n,6). If ACCROOT is
on, roots will be determined to a minimum of ROOTACC# significant
places. (If roots are closely spaced, a higher number of significant
places is computed where needed.)
:PREC: --(Default 8) This REDUCE parameter is used to determine the
precision of bigfloat computations. The function PRECISION n; causes
:PREC: to be set to the value n+2 but returns the value n. The roots
package, during its operation, will change the value of :PREC: but will
restore the original value of :PREC: at termination except that the
value of :PREC: is increased if necessary to allow the full output to be
printed.
ROOTPREC n; --The roots package normally sets the computation mode and
precision automatically if AUTOMODE is on. However, if ROOTPREC n; is
called and n>!!NFPD (where !!NFPD is the number of floating point
digits in the Lisp system,) then all root computation will be done
initially in bigfloat mode of minimum precision n. Automatic operation
can be restored by input of ROOTPREC 0;.
-----------------------------------------------------------------------
Avoiding truncation of polynomials on input;
The roots package will not internally truncate polynomials provided that
the switch automode is on (or, if automode is off, provided that
rootprec is not set to some value smaller than the number of significant
figures needed to represent the polynomial precisely.) However, it is
possible that a polynomial can be truncated by input reading functions
of the embedding lisp system, particularly when input is given in
floating point or bigfloat formats. (some lisp systems use the floating
point input routines to input bigfloats.)
To avoid any difficulties, input can be done in integer or Gaussian
integer format, or mixed, with integers or rationals used to represent
quantities of high precision. There are many examples of this in the
test package. Note that use of bigfloat of high precision will not
necessarily avoid truncation of coefficients if floating point input
format is used. It is usually sufficient to let the roots package
determine the precision needed to compute roots.
The number of digits that can be safely represented in floating point in
the lisp system are contained in the global variable !nfpd. Similarly,
the maximum number of significant figures in floating point output are
contained in the global variable !flim. The roots package computes
these values, which are needed to control the logic of the program.
The values of intermediate root iterations (that are printed when trroot
is on) are given in bigfloat format even when the actual values are
computed in floating point. This avoids intrusive rounding of root
printout.