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;; Polynomials
;; Liam Healy, Tue Mar 21 2006 - 18:33
;; Time-stamp: <2011-01-10 10:29:48EST polynomial.lisp>
;; Distributed under the terms of the GNU General Public License
;;
;; This program is free software: you can redistribute it and/or modify
;; it under the terms of the GNU General Public License as published by
;; the Free Software Foundation, either version 3 of the License, or
;; (at your option) any later version.
;;
;; This program is distributed in the hope that it will be useful,
;; but WITHOUT ANY WARRANTY; without even the implied warranty of
;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
;; GNU General Public License for more details.
;;
;; You should have received a copy of the GNU General Public License
;; along with this program. If not, see <http://www.gnu.org/licenses/>.
(in-package :gsl)
Liam Healy
committed
;;; /usr/include/gsl/gsl_poly.h
;;;;****************************************************************************
;;;; Polynomial Evaluation
;;;;****************************************************************************
((coefficients grid:vector-double-float) (x float) &key divided-difference)
((((foreign-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
(((foreign-pointer divided-difference) :pointer)
((foreign-pointer coefficients) :pointer)
((dim0 coefficients) sizet)
(x :double)))
:definition :method
:inputs (coefficients divided-difference)
:documentation ; FDL
"Evaluate the polyonomial with coefficients at the point x.")
((coefficients grid:vector-double-float) (x complex)
(((foreign-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
(x grid:complex-double-c))
:definition :method
:gsl-version (1 11)
:inputs (coefficients)
:c-return grid:complex-double-c
:documentation ; FDL
"Evaluate the polyonomial with coefficients at the complex value x.")
((coefficients grid:vector-complex-double-float) (x complex)
(((foreign-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
(x grid:complex-double-c))
:definition :method
:gsl-version (1 11)
:inputs (coefficients)
:c-return grid:complex-double-c
:documentation ; FDL
"Evaluate the polyonomial with coefficients at the complex value x.")
;;;;****************************************************************************
;;;; Divided Difference Representation of Polynomials
;;;;****************************************************************************
(defmfun divided-difference
(xa ya &optional (dd (grid:make-foreign-array 'double-float :dimensions (dim0 xa))))
"gsl_poly_dd_init"
(((foreign-pointer dd) :pointer)
((foreign-pointer xa) :pointer) ((foreign-pointer ya) :pointer)
((dim0 xa) sizet))
:inputs (xa ya)
:outputs (dd)
:return (dd)
:documentation ; FDL
"Compute a divided-difference representation of the
interpolating polynomial for the points (xa, ya) stored in
the arrays of equal length. On output the
divided-differences of (xa,ya) are stored in the array
dd, of the same length.")
(defmfun taylor-divided-difference
(xp dd xa
&optional
(coefficients (grid:make-foreign-array 'double-float :dimensions (dim0 xa)))
(workspace (grid:make-foreign-array 'double-float :dimensions (dim0 xa))))
"gsl_poly_dd_taylor"
(((foreign-pointer coefficients) :pointer)
((foreign-pointer dd) :pointer)
((foreign-pointer xa) :pointer)
((dim0 xa) sizet)
((foreign-pointer workspace) :pointer))
:inputs (dd xa)
:outputs (coefficients)
:documentation ; FDL
"Convert the divided-difference representation of a
polynomial to a Taylor expansion. The divided-difference representation
is supplied in the arrays dd and xa of the same length.
On output the Taylor coefficients of the polynomial expanded about the
point xp are stored in the array coefficients which has the same length
as xa and dd.")
;;;;****************************************************************************
;;;; Quadratic Equations
;;;;****************************************************************************
"gsl_poly_solve_quadratic"
((a :double) (b :double) (c :double)
(root1 (:pointer :double)) (root2 (:pointer :double)))
:c-return :number-of-answers
:documentation ; FDL
"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.")
"gsl_poly_complex_solve_quadratic"
((a :double) (b :double) (c :double)
(root1 (:pointer grid:complex-double-c)) (root2 (:pointer grid:complex-double-c)))
:c-return :number-of-answers
:documentation ; FDL
"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.")
;;;;****************************************************************************
;;;; Cubic Equations
;;;;****************************************************************************
"gsl_poly_solve_cubic"
((a :double) (b :double) (c :double)
(root1 (:pointer :double))
(root2 (:pointer :double)) (root3 (:pointer :double)))
:c-return :number-of-answers
:documentation ; FDL
"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.")
"gsl_poly_complex_solve_cubic"
((a :double) (b :double) (c :double)
(root1 (:pointer grid:complex-double-c)) (root2 (:pointer grid:complex-double-c))
(root3 (:pointer grid:complex-double-c)))
:c-return :number-of-answers
:documentation ; FDL
"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.")
;;;;****************************************************************************
;;;; General Polynomial Equations
;;;;****************************************************************************
(defmobject polynomial-complex-workspace "gsl_poly_complex_workspace"
((n sizet))
"complex workspace for polynomials")
(defmfun polynomial-solve
(coefficients
&optional
(answer (grid:make-foreign-array '(complex double-float)
:dimensions (1- (size coefficients))))
(workspace (make-polynomial-complex-workspace (size coefficients))))
"gsl_poly_complex_solve"
(((foreign-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
((mpointer workspace) :pointer) ((foreign-pointer answer) :pointer))
:outputs (answer)
:return (answer)
:documentation ; FDL
"Arguments are: a vector-double-float of coefficients, a complex
vector of length one less than coefficients that will hold the
answer, and a workspace made by make-polynomial-complex-workspace.
The roots of the general polynomial
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 coefficient of the
highest order term must be non-zero.")
;;;;****************************************************************************
;;;; Examples and unit test
;;;;****************************************************************************
(let* ((xa #m(0.0d0 1.0d0 2.0d0 3.0d0))
(dd (divided-difference xa #m(2.5d0 7.2d0 32.7d0 91.0d0))))
(evaluate xa 0.0d0 :divided-difference dd)
(evaluate xa 1.0d0 :divided-difference dd)
(evaluate xa 2.0d0 :divided-difference dd)
(evaluate xa 3.0d0 :divided-difference dd)))
Liam Healy
committed
(let ((vec #m(1.0d0 2.0d0 3.0d0)))
(solve-quadratic 1.0d0 0.0d0 1.0d0)
(solve-quadratic 1.0d0 -2.0d0 1.0d0)
(solve-quadratic-complex 1.0d0 -2.0d0 1.0d0)
(solve-cubic -6.0d0 -13.0d0 42.0d0)
(solve-cubic-complex -1.0d0 1.0d0 -1.0d0)
;; Example from GSL manual
(grid:copy-to (polynomial-solve #m(-1.0d0 0.0d0 0.0d0 0.0d0 0.0d0 1.0d0)))
;; tests from gsl-1.11/poly/test.c
(evaluate #m(1.0d0 0.5d0 0.3d0) 0.5d0)
(evaluate
#m(1.0d0 -1.0d0 1.0d0 -1.0d0 1.0d0 -1.0d0 1.0d0 -1.0d0 1.0d0 -1.0d0 1.0d0)
1.0d0)
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(solve-quadratic 4.0d0 -20.0d0 26.0d0) ; no roots
(solve-quadratic 4.0d0 -20.0d0 25.0d0) ; one root
(solve-quadratic 4.0d0 -20.0d0 21.0d0) ; two roots
(solve-quadratic 4.0d0 7.0d0 0.0d0) ; two roots
(solve-quadratic 5.0d0 0.0d0 -20.0d0) ; two roots
(solve-quadratic 0.0d0 3.0d0 -21.0d0) ; one root (linear)
(solve-quadratic 0.0d0 0.0d0 1.0d0)
(solve-cubic 0.0d0 0.0d0 -27.0d0) ; one root
(solve-cubic -51.0d0 867.0d0 -4913.0d0) ; three roots
(solve-cubic -57.0d0 1071.0d0 -6647.0d0) ; three roots
(solve-cubic -11.0d0 -493.0d0 +6647.0d0) ; three roots
(solve-cubic -143.0d0 5087.0d0 -50065.0d0) ; three roots
(solve-cubic -109.0d0 803.0d0 50065.0d0) ; three roots
(solve-quadratic-complex 4.0d0 -20.0d0 26.0d0)
(solve-quadratic-complex 4.0d0 -20.0d0 25.0d0)
(solve-quadratic-complex 4.0d0 -20.0d0 21.0d0)
(solve-quadratic-complex 4.0d0 7.0d0 0.0d0)
(solve-quadratic-complex 5.0d0 0.0d0 -20.0d0)
(solve-quadratic-complex 5.0d0 0.0d0 20.0d0)
(solve-quadratic-complex 0.0d0 3.0d0 -21.0d0)
(solve-quadratic-complex 0.0d0 0.0d0 1.0d0)
(solve-cubic-complex 0.0d0 0.0d0 -27.0d0)
(solve-cubic-complex -1.0d0 1.0d0 39.0d0)
(solve-cubic-complex -51.0d0 867.0d0 -4913.0d0)
(solve-cubic-complex -57.0d0 1071.0d0 -6647.0d0)
(solve-cubic-complex -11.0d0 -493.0d0 +6647.0d0)
(solve-cubic-complex -143.0d0 5087.0d0 -50065.0d0)
(grid:copy-to (polynomial-solve #m(-120.0d0 274.0d0 -225.0d0 85.0d0 -15.0d0 1.0d0)))
(grid:copy-to (polynomial-solve #m(1.0d0 0.0d0 0.0d0 0.0d0 1.0d0 0.0d0 0.0d0 0.0d0 1.0d0)))
(let* ((xa #m(0.16d0 0.97d0 1.94d0 2.74d0 3.58d0 3.73d0 4.70d0))
(ya #m(0.73d0 1.11d0 1.49d0 1.84d0 2.30d0 2.41d0 3.07d0))
(dd (divided-difference xa ya)))
(list
(map 'vector (lambda (x) (evaluate xa x :divided-difference dd))
(map 'vector
(lambda (x) (evaluate (taylor-divided-difference 1.5d0 dd xa) (- x 1.5d0)))