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;********************************************************
; file: polynomial.lisp
; description: Polynomials
; date: Tue Mar 21 2006 - 18:33
; author: Liam M. Healy
; modified: Fri Apr 28 2006 - 22:27
;********************************************************
;;; $Id: $
(in-package :gsl)
;;; To do:
;;; Divided difference needs to be checked.
;;; polynomial-solve-ws awaits complex vectors
;;;;****************************************************************************
;;;; Polynomial Evaluation
;;;;****************************************************************************
;;; (defparameter vec (make-data 'vector nil 3))
;;; (setf (data vec) #(1.0d0 2.0d0 3.0d0))
;;; (polynomial-eval vec -1.0d0)
;;; 2.0d0
(defun-gsl polynomial-eval
(((gsl-array coefficients) :pointer) ((dim0 coefficients) :size) (x :double))
"gsl_poly_eval"
:function (coefficients x)
:documentation
"Evaluate the polyonomial with coefficients at the point x."
:return (:double)
:c-return-value :return)
;;;;****************************************************************************
;;;; Divided Difference Representation of Polynomials
;;;;****************************************************************************
;;; Use with-divided-difference to compute the divided difference, an
;;; opaque object that is then passed to
;;; get-divided-difference, eval-divided-difference or taylor-divided-difference
;;; in the body.
(export '(with-divided-difference))
(defmacro with-divided-difference ((dd xa ya) &body body)
"Compute the divided difference and bind dd to it.
This variable may then be passed to divided-difference
functions in the body."
`(let ((,dd (find-divided-difference ,xa ,ya)))
(unwind-protect
(progn ,@body)
(cffi::foreign-array-free (first ,dd))
(cffi::foreign-array-free (second ,dd)))))
;;; Do not call this function directly; use with-divided-difference.
(defun find-divided-difference (xa ya)
"Find the divided difference representation of the interpolating polynomial
for the points stored in the arrays @var{xa} and @var{ya}."
(let* ((size (length xa))
(dd (foreign-alloc :double :count size))
(xac (foreign-alloc :double :count size)))
(cffi::with-foreign-array (yac ya :double (list size))
;; return code not checked
(cffi:foreign-funcall
"gsl_poly_dd_init"
:pointer dd
:pointer xac
:pointer yac
:int))
(list dd xac size)))
(defunx get-divided-difference (dd)
"Convert the divided difference into an array."
(cffi::foreign-array-to-lisp
(first dd) :double (list (third dd))))
(defunx-map eval-divided-difference "gsl_poly_dd_eval" (dd x)
"Evaluate the polynomial stored in divided-difference form
at the point @var{x}. Call only within a
with-divided-difference form."
(cffi:foreign-funcall
"gsl_poly_dd_eval"
:pointer (first dd)
:pointer (second dd)
:size (third dd)
:double x
:double))
(defunx-map taylor-divided-difference "gsl_poly_dd_taylor" (dd xp)
"Convert the divided-difference representation of a polynomial
to a Taylor expansion about the point xp. Call only within a
with-divided-difference form."
(let ((cc (foreign-alloc :double :count (third dd)))
(workspace (foreign-alloc :double :count (third dd))))
(unwind-protect
(progn
;; Return value not checked.
(cffi:foreign-funcall
"gsl_poly_dd_taylor"
:pointer cc
:double xp
:pointer (first dd)
:pointer (second dd)
:size (third dd)
:pointer workspace
:int)
(cffi::foreign-array-to-lisp
cc :double (list (third dd))))
(cffi::foreign-array-free workspace)
(cffi::foreign-array-free cc))))
;;;;****************************************************************************
;;;; Quadratic Equations
;;;;****************************************************************************
(defun-gsl solve-quadratic ((a :double) (b :double) (c :double))
"gsl_poly_solve_quadratic"
:documentation
"The real roots of the quadratic equation a x^2 + b x + c = 0.
Two values are always returned; if the roots are not real, these
values are NIL."
:return (:double :double)
:c-return-value :number-of-answers)
(defun-gsl solve-quadratic-complex ((a :double) (b :double) (c :double))
"gsl_poly_complex_solve_quadratic"
:documentation
"The complex roots of the quadratic equation a x^2 + b x + c = 0.
Two values are always returned; if a root does not exist, the
value returned will be NIL."
:return (gsl-complex gsl-complex)
:c-return-value :number-of-answers)
;;;;****************************************************************************
;;;; Cubic Equations
;;;;****************************************************************************
;;; (solve-cubic -6.0d0 -13.0d0 42.0d0)
;;; -3.0d0
;;; 1.9999999999999996d0
;;; 7.0d0
;;; (solve-cubic -1.0d0 1.0d0 -1.0d0)
;;; 1.0d0
;;; NIL
;;; NIL
(defun-gsl solve-cubic ((a :double) (b :double) (c :double))
"gsl_poly_solve_cubic"
:documentation
"Find the real roots of the cubic equation, x^3 + a x^2 + b x + c = 0
with a leading coefficient of unity. The roots are given
in ascending order. Three values are always returned;
if a root is not real, the value returned for it will be NIL."
:return (:double :double :double)
:c-return-value :number-of-answers)
(defun-gsl solve-cubic-complex ((a :double) (b :double) (c :double))
"gsl_poly_complex_solve_cubic"
:documentation
"Find the complex roots of the cubic equation, x^3 + a x^2 + b x + c = 0
with a leading coefficient of unity. Three values are always returned;
if a root does not exist, the value returned for it will be NIL."
:return (gsl-complex gsl-complex gsl-complex)
:c-return-value :number-of-answers)
;;;;****************************************************************************
;;;; General Polynomial Equations
;;;;****************************************************************************
;;; Note:
;;; If one polynomial needs to be solved #'polynomial-solve
;;; without the second argument is appropriate. If multiple polynomials
;;; of the same degree need to be solved, then use with-poly-complex-workspace
;;; once, wrapping the multiple calls to polynomial-solve with an
;;; explicit workspace argument. This insures that the workspace, which
;;; GSL needs for scratch space, is allocated and deallocated only once.
#|
;;; Example from GSL manual:
(with-poly-complex-workspace (ws 6)
(polynomial-solve #(-1.0d0 0.0d0 0.0d0 0.0d0 0.0d0 1.0d0) ws))
(polynomial-solve #(-1.0d0 0.0d0 0.0d0 0.0d0 0.0d0 1.0d0))
(#C(-0.8090169943749475d0 0.5877852522924731d0)
#C(-0.8090169943749475d0 -0.5877852522924731d0)
#C(0.30901699437494745d0 0.9510565162951536d0)
#C(0.30901699437494745d0 -0.9510565162951536d0)
#C(1.0000000000000002d0 0.0d0))
|#
;;; See /usr/include/gsl/gsl_poly.h
(cffi:defcstruct poly-complex-workspace
(matrix :pointer))
(export '(with-poly-complex-workspace))
(defmacro with-poly-complex-workspace ((workspace size) &body body)
"Macro to create and cleanup workspace for polynomial root solver."
`(let ((,workspace
(funcall
"gsl_poly_complex_workspace_alloc"
:return
(poly-complex-workspace)
:c-return-value :return)
,size)))
(unwind-protect
(progn ,@body)
(funcall
(defun-gsl :lambda ((,workspace poly-complex-workspace))
"gsl_poly_complex_workspace_free"
:c-return-value :void)
,workspace))))
(defun-gsl polynomial-solve-ws
(((gsl-array coefficients) :pointer) ((dim0 coefficients) :size)
(workspace poly-complex-workspace))
"gsl_poly_complex_solve"
:function (coefficients workspace)
:return ((gsl-complex (1- (dim0 coefficients)))))
(export '(polynomial-solve))
(defmacro polynomial-solve (coefficients &optional workspace)
"The roots of the general polynomial
@c{$P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}$}
@math{P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_@{n-1@} x^@{n-1@}} using
balanced-QR reduction of the companion matrix. The parameter @var{n}
specifies the length of the coefficient array. The coefficient of the
highest order term must be non-zero. The function requires a workspace
@var{w} of the appropriate size. The @math{n-1} roots are returned in
the packed complex array @var{z} of length @math{2(n-1)}, alternating
real and imaginary parts."
(if workspace
`(polynomial-solve-ws ,coefficients ,workspace)
(let ((ws (gensym "WS"))
(coef (gensym "COEF")))
`(let ((,coef ,coefficients))
(with-poly-complex-workspace (,ws (length ,coef))
(polynomial-solve-ws ,coef ,ws))))))