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;; Author Barton Willis
;; University of Nebraska at Kearney
;; Copyright (C) 2011,2021 Barton Willis
;; 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 2 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.
(in-package :maxima)
($load "bernstein_utilities.mac")
;; When bernstein_explicit is non-nil, bernstein_poly(k,n,x) simplifies to
;; binomial(n,k) * x^k (1-x)^(n-k) regardless of the values of k or n;
(defmvar $bernstein_explicit nil)
;; numerical (complex rational, float, or big float) evaluation of bernstein polynomials
(in-package #:bigfloat)
(defun bernstein-poly (k n x)
(* (to (maxima::opcons 'maxima::%binomial n k)) (expt x k) (expt (- 1 x) (- n k))))
(in-package :maxima)
(defun $bernstein_poly (k n x) (simplify (list '(%bernstein_poly) k n x)))
(defprop $bernstein_poly %bernstein_poly alias)
(defprop $bernstein_poly %bernstein_poly verb)
(defprop %bernstein_poly $bernstein_poly reversealias)
(defprop %bernstein_poly $bernstein_poly noun)
(setf (get '%bernstein_poly 'conjugate-function)
#'(lambda (e)
(let ((k (car e))
(n (cadr e))
(x (caddr e)))
(if (and ($featurep k '$integer) ($featurep n '$integer))
(opcons '%bernstein_poly k n (opcons '$conjugate x))
(list (list '$conjugate 'simp) (take '(%bernstein_poly) k n x))))))
;; integrate(bernstein_poly(k,n,x),x) = hypergeometric([k+1,k-n],[k+2],x)*x^(k+1)/(k+1)
(defun bernstein-integral (k n x)
(div
(mul
(opcons '%binomial n k)
(opcons 'mexpt x (add 1 k))
(opcons '%hypergeometric
(opcons 'mlist (add 1 k) (sub k n))
(opcons 'mlist (add 2 k))
x))
(add 1 k)))
(putprop '%bernstein_poly `((k n x) nil nil ,'bernstein-integral) 'integral)
(putprop '$bernstein_poly `((k n x) nil nil ,'bernstein-integral) 'integral)
(defun bernstein-poly-simp (e y z)
(declare (ignore y))
(let* ((fn (car (pop e)))
(k (if (consp e) (simpcheck (pop e) z) (wna-err fn)))
(n (if (consp e) (simpcheck (pop e) z) (wna-err fn)))
(x (if (consp e) (simpcheck (pop e) z) (wna-err fn))))
(if (consp e) (wna-err fn))
(cond ((and (integerp k) (integerp n) (>= k 0) (>= n k)
(complex-number-p x #'(lambda (s) (or (integerp s) ($ratnump s) (floatp s) ($bfloatp s)))))
(maxima::to (bigfloat::bernstein-poly (bigfloat::to k) (bigfloat::to n) (bigfloat::to x))))
((zerop1 x) (opcons '%kron_delta k 0))
((onep1 x) (opcons '%kron_delta k n))
((or $bernstein_explicit (and (integerp k) (integerp n)))
(if (and (integerp k) (integerp n) (or (< k 0) (> k n))) (mul 0 x)
(mul (opcons '%binomial n k) (opcons 'mexpt x k) (opcons 'mexpt (sub 1 x) (sub n k)))))
(t (list (list fn 'simp) k n x)))))
(setf (get '%bernstein_poly 'operators) 'bernstein-poly-simp)
(defprop %bernstein_poly
((k n x)
((mtimes) ((%bernstein_poly) k n x)
((mplus)
((mtimes) -1
((mqapply) (($psi array) 0) ((mplus) 1 k)))
((mqapply) (($psi array) 0)
((mplus) 1 ((mtimes) -1 k) n))
((mtimes) -1
((%log) ((mplus) 1 ((mtimes) -1 x))))
((%log) x)))
((mtimes)((%bernstein_poly) k n x)
((mplus)
((mqapply) (($psi array) 0) ((mplus) 1 n))
((mtimes) -1
((mqapply) (($psi array) 0)
((mplus) 1 ((mtimes) -1 k) n)))
((%log) ((mplus) 1 ((mtimes) -1 x)))))
((mtimes)
((mplus)
((%bernstein_poly) ((mplus) -1 k) ((mplus) -1 n) x)
((mtimes) -1
((%bernstein_poly) k ((mplus) -1 n) x))) n))
grad)
(defun $bernstein_approx (e vars n)
(if (not ($listp vars)) (merror "The second argument to bernstein_approx must be a list"))
(setq vars (margs vars))
(if (some #'(lambda (s) (not ($mapatom s))) vars)
(merror "The second argument to bernstein_approx must be a list of atoms"))
(if (or (not (integerp n)) (< n 1)) (merror "The third argument to bernstein_approx must be a positive integer"))
(let* ((k (length vars))
(d (make-list k :initial-element 0))
(nn (make-list k :initial-element n))
(carry) (acc 0) (m) (x))
(setq m (expt (+ n 1) k))
(setq nn (cons '(mlist) nn))
(dotimes (i m)
(setq acc (add acc
(mul
(opcons '%multibernstein_poly (cons '(mlist) d) nn (cons '(mlist) vars))
($substitute (cons '(mlist) (mapcar #'(lambda (s x) (opcons 'mequal x (div s n))) d vars)) e))))
(setq carry 1)
(dotimes (j k)
(setq x (+ carry (nth j d)))
(if (> x n) (setq x 0 carry 1) (setq carry 0))
(setf (nth j d) x)))
acc))
(defun $multibernstein_poly (k n x) (simplify (list '(%multibernstein_poly) k n x)))
(defprop $multibernstein_poly %multibernstein_poly alias)
(defprop $multibernstein_poly %multibernstein_poly verb)
(defprop %multibernstein_poly $multibernstein_poly reversealias)
(defprop %multibernstein_poly $multibernstein_poly noun)
(defun multi-bernstein-poly-simp (e y z)
(declare (ignore y))
(let*
((fn (car (pop e)))
(k (if (consp e) (simpcheck (pop e) z) (wna-err fn)))
(n (if (consp e) (simpcheck (pop e) z) (wna-err fn)))
(x (if (consp e) (simpcheck (pop e) z) (wna-err fn))))
(if (consp e) (wna-err fn))
(if (or (not (and ($listp k) ($listp n) ($listp x))) (/= ($length k) ($length n)) (/= ($length n) ($length x)))
(merror "Each argument to multibernstein_poly must be an equal length list"))
(muln (mapcar #'(lambda (a b z) (opcons '%bernstein_poly a b z)) (margs k) (margs n) (margs x)) t)))
(setf (get '%multibernstein_poly 'operators) 'multi-bernstein-poly-simp)
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