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;; Polynomials
;; Liam Healy, Tue Mar 21 2006 - 18:33
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;; Time-stamp: <2009-01-31 19:34:56EST polynomial.lisp>
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;; $Id$
;;; Provide autotranslation from CL pure arrays?
;;; Divided differences not complete/tested.

;;;;****************************************************************************
;;;; Polynomial Evaluation
;;;;****************************************************************************

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(defmfun evaluate
    ((coefficients vector-double-float) (x float) &key divided-difference)
  ("gsl_poly_eval" "gsl_poly_dd_eval")
  ((((c-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
    (x :double))
   (((c-pointer divided-difference) :pointer)
    ((c-pointer coefficients) :pointer)
    ((dim0 coefficients) sizet)
    (x :double)))
  :definition :method
  :inputs (coefficients divided-difference)
  :c-return :double
  "Evaluate the polyonomial with coefficients at the point x.")
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#|
;;; These won't work until we can take returned complex (structs).
(defmfun evaluate
    ((coefficients vector-double-float) (x complex)
     &key &allow-other-keys)
  "gsl_poly_complex_eval"
  (((c-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
   (x complex-double-c))
  :definition :method
  :gsl-version (1 11)
  :inputs (coefficients)
  :c-return complex-double-c
  :documentation			; FDL
  "Evaluate the polyonomial with coefficients at the complex value x.")

(defmfun evaluate
    ((coefficients vector-complex-double-float) (x complex)
     &key &allow-other-keys)
  "gsl_complex_poly_complex_eval"
  (((c-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
   (x complex-double-c))
  :definition :method
  :gsl-version (1 11)
  :inputs (coefficients)
  :c-return complex-double-c
  :documentation			; FDL
  "Evaluate the polyonomial with coefficients at the complex value x.")
|#

;;;;****************************************************************************
;;;; Divided Difference Representation of Polynomials
;;;;****************************************************************************

(defmfun divided-difference (dd xa ya)
  (((c-pointer dd) :pointer)
   ((c-pointer xa) :pointer) ((c-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
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(defmfun taylor-divided-difference (coefs xp dd xa workspace)
  (((c-pointer coefs) :pointer)
   ((c-pointer dd) :pointer)
   ((c-pointer xa) :pointer)
   ((dim0 xa) sizet)
   ((c-pointer workspace) :pointer))
  :inputs (coefs xa)
  :outputs (coefs)
  "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 coefs which has the same length
  as xa and dd.  A workspace of that length must be provided.")

;;;;****************************************************************************
;;;; Quadratic Equations
;;;;****************************************************************************

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(defmfun solve-quadratic (a b c)
  "gsl_poly_solve_quadratic"
  ((a :double) (b :double) (c :double) (root1 :double) (root2 :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
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(defmfun solve-quadratic-complex (a b c)
  "gsl_poly_complex_solve_quadratic"
  ((a :double) (b :double) (c :double)
   (root1 complex-double-c) (root2 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
;;;;****************************************************************************

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(defmfun solve-cubic (a b c)
  "gsl_poly_solve_cubic"
  ((a :double) (b :double) (c :double)
   (root1 :double) (root2 :double) (root3 :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.")
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(defmfun solve-cubic-complex (a b c)
  "gsl_poly_complex_solve_cubic"
  ((a :double) (b :double) (c :double)
   (root1 complex-double-c) (root2 complex-double-c) (root3 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 (make-marray '(complex double-float)
			  :dimensions (1- (total-size coefficients))))
     (workspace (make-polynomial-complex-workspace (total-size coefficients))))
  (((c-pointer coefficients) :pointer) ((dim0 coefficients) sizet)
   ((mpointer workspace) :pointer) ((c-pointer answer) :pointer))
  :inputs (coefficients)
  :outputs (answer)
  :return (answer)
  "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
;;;;****************************************************************************

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(save-test polynomial
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       (ya #m(2.5d0 7.2d0 32.7d0 91.0d0))
       (dd (make-marray 'double-float :dimensions 4)))
   (divided-difference dd xa ya)
   (list
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    (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)))
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   (evaluate vec -1.0d0))
 (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
 (cl-array (polynomial-solve #m(-1.0d0 0.0d0 0.0d0 0.0d0 0.0d0 1.0d0))))