optimize.lisp

   1 ;;; -*- mode: lisp -*-
   2 ;;; Copyright (c) 2005--2007, by A.J. Rossini <blindglobe@gmail.com>
   3 ;;; See COPYRIGHT file for any additional restrictions (BSD license).
   4 ;;; Since 1991, ANSI was finally finished.  Edited for ANSI Common Lisp.
   5
   6 (in-package :cl-user)
   7
   8 (defpackage :lisp-stat-optimize
   9  (:use :common-lisp
  10        :lisp-stat-ffi-int
  11        :lisp-stat-object-system
  12        :lisp-stat-types
  13        :lisp-stat-compound-data
  14        :lisp-stat-math
  15        :lisp-stat-float
  16        :lisp-stat-basics
  17        :lisp-stat-matrix
  18        :lisp-stat-linalg-data
  19        :lisp-stat-linalg)
  20  (:shadowing-import-from :lisp-stat-object-system
  21                          slot-value call-method call-next-method)
  22  (:shadowing-import-from :lisp-stat-math
  23            expt + - * / ** mod rem abs 1+ 1- log exp sqrt sin cos tan
  24            asin acos atan sinh cosh tanh asinh acosh atanh float random
  25            truncate floor ceiling round minusp zerop plusp evenp oddp
  26            < <= = /= >= > complex conjugate realpart imagpart phase
  27            min max logand logior logxor lognot ffloor fceiling
  28            ftruncate fround signum cis)
  29  (:export
  30      ;; derivatives
  31      numgrad numhess
  32
  33      ;; optimization
  34      newtonmax nelmeadmax))
  35
  36 ;; matrix is in statistics, but should that be a predecessor?
  37
  38 ;;; FIXME:AJR: There is a need to figure out the proper symbols to
  39 ;;; export.  more importantly should there be any specialty package
  40 ;;; that are exported for maximization?
  41
  42 (in-package :lisp-stat-optimize)
  43
  44 (defvar *maximize-callback-function* nil
  45   "Used in generic optimization to determine function name -- symbol or string?")
  46
  47 (defvar *maximize-callback-arg* nil
  48   "args to function to maximize")
  49
  50
  51 ;;;
  52 ;;; CFFI support using library for optimization work.
  53 ;;;
  54
  55 ;; There is a problem with this particular approach, in terms of
  56 ;; circular dependencies.  We can not have this out-of-object call
  57 ;; into optimize, at least not from here.
  58 (cffi:defcallback ccl-maximize-callback :void ((n :int)
  59                                                (px :pointer)
  60                                                (pfval :pointer)
  61                                                (pgrad :pointer)
  62                                                (phess :pointer)
  63                                                (pderivs :pointer))
  64    (lisp-stat-optimize::maximize-callback n px pfval pgrad phess pderivs))
  65
  66 (cffi:defcfun ("register_maximize_callback" register-maximize-callback)
  67     :void (x :pointer))
  68 (register-maximize-callback (cffi:callback ccl-maximize-callback))
  69
  70 (cffi:defcfun ("ccl_numgrad_front" ccl-numgrad-front)
  71     :int (x :int) (y :pointer) (z :pointer) (u :double) (v :pointer))
  72 (defun numgrad-front (x y z u v)
  73   (ccl-numgrad-front x y z (float u 1d0) v))
  74
  75 (cffi:defcfun ("ccl_numhess_front" ccl-numhess-front)
  76     :int (x :int) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :double) (a :pointer))
  77 (defun numhess-front (x y z u v w a)
  78   (ccl-numhess-front x y z u v (float w 1d0) a))
  79
  80 (cffi:defcfun ("ccl_minfo_maximize" ccl-minfo-maximize)
  81     :int (x :pointer) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :int))
  82 (defun base-minfo-maximize (x y z u v w)
  83   (ccl-minfo-maximize x y z u v w))
  84
  85
  86
  87 ;;;;
  88 ;;;; minfo basics (internal??)
  89 ;;;;
  90
  91 (defun init-minfo-ipar-values (n ipars &key
  92                                (TRUE 1)
  93                                (FALSE 0)
  94                                (k 0)
  95                                (m 0)
  96                                (itnlimit -1)
  97                                (backtrack TRUE)
  98                                (verbose 0)
  99                                (vals_suppl FALSE)
 100                                (exptilt TRUE)
 101                                (count 0)
 102                                (termcode 0))
 103   "Initialize ipars (iteration parameters) by destructive modification."
 104   (setf (aref ipars 0) n)
 105   (setf (aref ipars 1) m)
 106   (setf (aref ipars 2) k)
 107   (setf (aref ipars 3) itnlimit)
 108   (setf (aref ipars 4) backtrack)
 109   (setf (aref ipars 5) verbose)
 110   (setf (aref ipars 6) vals_suppl)
 111   (setf (aref ipars 7) exptilt)
 112   (setf (aref ipars 8) count)
 113   (setf (aref ipars 9) termcode))
 114
 115 (defun init-minfo-dpar-values (h dpars &key
 116                                (typf 1.0)
 117                                (gradtol -1.0)
 118                                (steptol -1.0)
 119                                (maxstep -1.0)
 120                                (dflt 0.0)
 121                                (tilt 0.0)
 122                                (newtilt 0.0)
 123                                (hessadd 0.0))
 124   "Initialize dpars (derivative parameters) by destructive modification."
 125   (setf (aref dpars 0) typf)
 126   (setf (aref dpars 1) h)
 127   (setf (aref dpars 2) gradtol)
 128   (setf (aref dpars 3) steptol)
 129   (setf (aref dpars 4) maxstep)
 130   (setf (aref dpars 5) dflt)
 131   (setf (aref dpars 6) tilt)
 132   (setf (aref dpars 7) newtilt)
 133   (setf (aref dpars 8) hessadd))
 134
 135 (defun init-minfo-internals (n h internals)
 136   (let ((ipars (aref internals 8))
 137         (dpars (aref internals 9)))
 138     (init-minfo-ipar-values n ipars)
 139     (init-minfo-dpar-values h dpars)))
 140
 141 (defun new-minfo-internals (f x &key scale ((:derivstep h) -1.0))
 142   (check-sequence x)
 143   (check-real x)
 144   (check-one-real h)
 145   (let ((n (length x)))
 146     (when scale
 147           (check-sequence scale)
 148           (check-real scale)
 149           (if (/= n (length scale)) (error "scale and x not the same length")))
 150     (let ((internals (make-array 12)))
 151       (setf (aref internals 0) f)
 152       (setf (aref internals 3) (if (consp x) (copy-list x) (coerce x 'list)))
 153       (setf (aref internals 4)
 154             (if scale (copy-seq scale) (make-array n :initial-element 1.0)))
 155       (setf (aref internals 5) (make-list (+ 1 n (* n n))))
 156       (setf (aref internals 8) (make-array 10))
 157       (setf (aref internals 9) (make-array 9))
 158       (init-minfo-internals n h internals)
 159       internals)))
 160
 161 (defun minfo-maximize (internals &optional verbose)
 162   "This function does what?"
 163   (let* ((f (aref internals 0))
 164          (x (aref internals 3))
 165          (fvals (aref internals 5))
 166          (n (length x))
 167          (v (if verbose (if (integerp verbose) verbose 1) -1)))
 168     (setf (aref internals 3) (copy-list x))
 169     (setf (aref internals 5) (copy-list fvals))
 170     (let ((*maximize-callback-function* f)
 171           (*maximize-callback-arg* (make-list n)))
 172       (let* ((x (aref internals 3))
 173              (scale (aref internals 4))
 174              (fvals (aref internals 5))
 175              (ip (aref internals 8))
 176              (dp (aref internals 9))
 177              (px (la-data-to-vector x +mode-re+))
 178              (pscale (la-data-to-vector scale +mode-re+))
 179              (pfvals (la-vector (length fvals) +mode-re+))
 180              (pip (la-data-to-vector ip +mode-in+))
 181              (pdp (la-data-to-vector dp +mode-re+)))
 182         (unwind-protect
 183             (progn
 184               (base-minfo-maximize px pfvals pscale pip pdp v)) ;; access to C
 185           (la-vector-to-data px n +mode-re+ x)
 186           (la-vector-to-data pfvals (+ 1 n (* n n)) +mode-re+ fvals)
 187           (la-vector-to-data pip (length ip) +mode-in+ ip)
 188           (la-vector-to-data pdp (length dp) +mode-re+ dp))
 189         (get-buf)))))
 190
 191
 192
 193 ;;;;
 194 ;;;; Mode Info Prototype
 195 ;;;;
 196
 197 (defvar minfo-proto)
 198 (defproto minfo-proto '(internals))
 199
 200 #+xlisp (send minfo-proto :add-method :isnew #'|minfo-isnew|)
 201 #+xlisp (send minfo-proto :add-method :maximize #'|minfo-maximize|)
 202 #+xlisp (send minfo-proto :add-method :loglaplace #'|minfo-loglap|)
 203 #-xlisp
 204 (defmeth minfo-proto :isnew (&rest args)
 205   (setf (slot-value 'internals) (apply #'new-minfo-internals args)))
 206 #-xlisp
 207 (defmeth minfo-proto :maximize (&rest args)
 208   (apply #'minfo-maximize (slot-value 'internals) args))
 209
 210 (defmeth minfo-proto :x () (aref (slot-value 'internals) 3))
 211 (defmeth minfo-proto :scale () (aref (slot-value 'internals) 4))
 212 (defmeth minfo-proto :derivstep () (aref (aref (slot-value 'internals) 9) 1))
 213 (defmeth minfo-proto :tilt () (aref (aref (slot-value 'internals) 9) 6))
 214
 215 (defmeth minfo-proto :f (&optional (val nil set))
 216   (when set
 217         (send self :set-no-vals-supplied)
 218         (setf (aref (slot-value 'internals) 0) val))
 219   (aref (slot-value 'internals) 0))
 220
 221 (defmeth minfo-proto :set-no-vals-supplied ()
 222   (setf (aref (aref (slot-value 'internals) 8) 6) 0))
 223
 224 (defmeth minfo-proto :exptilt (&optional (val nil set))
 225   (if set
 226       (let ((old (send self :exptilt)))
 227         (setf (aref (aref (slot-value 'internals) 8) 7) (if val 1 0))
 228         (if (and (not (or (and old val) (and (not old) (not val))))
 229                  (/= (send self :tilt) 0.0))
 230             (send self :set-no-vals-supplied))))
 231   (= 1 (aref (aref (slot-value 'internals) 8) 7)))
 232
 233 (defmeth minfo-proto :newtilt (&optional (val nil set))
 234   (when set
 235         (setf (aref (aref (slot-value 'internals) 9) 7) (float val))
 236         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 237   (aref (aref (slot-value 'internals) 9) 7))
 238
 239 (defmeth minfo-proto :gfuns (&optional (val nil set))
 240   (when set
 241         (if (or (not (consp val))
 242                 (not (every #'functionp val)))
 243             (error "not all functions"))
 244         (setf (aref (slot-value 'internals) 1) val)
 245         (setf (aref (aref (slot-value 'internals) 8) 1) (length val))
 246         (setf (aref (slot-value 'internals) 10) (repeat 1.0 (length val)))
 247         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 248   (aref (slot-value 'internals) 1))
 249
 250 (defmeth minfo-proto :cfuns (&optional (val nil set))
 251   (when set
 252         (if (or (not (consp val))
 253                 (not (every #'functionp val)))
 254             (error "not all functions"))
 255         (setf (aref (slot-value 'internals) 2) val)
 256         (setf (aref (aref (slot-value 'internals) 8) 2) (length val))
 257         (setf (aref (slot-value 'internals) 7) (repeat 0.0 (length val)))
 258         (setf (aref (slot-value 'internals) 11) (repeat 0.0 (length val)))
 259         (send self :set-no-vals-supplied))
 260   (aref (slot-value 'internals) 2))
 261
 262 (defmeth minfo-proto :ctarget (&optional (val nil set))
 263   (when set
 264         (if (/= (length val) (length (send self :ctarget)))
 265             (error "bad target length"))
 266         (setf (aref (slot-value 'internals) 7) val))
 267   (aref (slot-value 'internals) 7))
 268
 269 (defmeth minfo-proto :fvals ()
 270   (let* ((fv (aref (slot-value 'internals) 5))
 271          (n (length (send self :x)))
 272          (val (select fv 0))
 273          (grad (select fv (iseq 1 n)))
 274          (hess (matrix (list n n) (select fv (iseq (+ 1 n) (+ n (* n n)))))))
 275     (list val grad hess)))
 276
 277 (defmeth minfo-proto :copy ()
 278   (let ((obj (make-object minfo-proto))
 279         (internals (copy-seq (slot-value 'internals))))
 280     (dotimes (i (length internals))
 281              (let ((x (aref internals i)))
 282                (if (sequencep x)
 283                    (setf (aref internals i) (copy-seq x)))))
 284     (send obj :add-slot 'internals internals)
 285     obj))
 286
 287 (defmeth minfo-proto :derivscale ()
 288   (let* ((step (^ machine-epsilon (/ 1 6)))
 289          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 290          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 291     (setf hess (numhess (send self :f) (send self :x) scale step))
 292     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 293     (setf (aref (slot-value 'internals) 4) scale)
 294     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 295
 296 (defmeth minfo-proto :verbose (&optional (val nil set))
 297   (when set
 298         (setf (aref (aref (slot-value 'internals) 8) 5)
 299               (cond ((integerp val) val)
 300                     ((null val) 0)
 301                     (t 1))))
 302   (aref (aref (slot-value 'internals) 8) 5))
 303
 304 (defmeth minfo-proto :backtrack (&optional (val nil set))
 305   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 306   (aref (aref (slot-value 'internals) 8) 4))
 307
 308 (defmeth minfo-proto :maxiter (&optional (val nil set))
 309     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 310                   (if (integerp val) val -1)))
 311     (aref (aref (slot-value 'internals) 8) 3))
 312
 313 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 314   (when set
 315         (if (/= (length val) (length (send self :gfuns)))
 316             (error "wrong size tilt scale sequence"))
 317         (setf (aref (slot-value 'internals) 10) val))
 318   (aref (slot-value 'internals) 10))
 319
 320 ;;;;
 321 ;;;;
 322 ;;;; Newton's Method with Backtracking
 323 ;;;;
 324 ;;;;
 325
 326 (defun newtonmax (f start &key
 327                     scale
 328                     (derivstep -1.0)
 329                     (count-limit -1)
 330                     (verbose 1)
 331                     return-derivs)
 332 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 333 Maximizes F starting from START using Newton's method with backtracking.
 334 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 335 list of location, unction value, gradient and hessian at maximum.
 336 SCALE should be a list of the typical magnitudes of the parameters.
 337 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 338 of iteration information. COUNT-LIMIT limits the number of iterations"
 339   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 340         (minfo (send minfo-proto :new f start
 341                      :scale scale :derivstep derivstep)))
 342     (send minfo :maxiter count-limit)
 343     (send minfo :derivscale)
 344     (send minfo :maximize verbose)
 345     (if return-derivs
 346         (cons (send minfo :x) (- (send minfo :fvals)))
 347         (send minfo :x))))
 348
 349 ;;;
 350 ;;; Nelder-Mead Simplex Method
 351 ;;;
 352
 353 ;;; Simplex Prototype
 354
 355 (defvar simplex-proto)
 356 (defproto simplex-proto '(f simplex))
 357
 358 (defun nelmeadmax (f start &key
 359                      (size 1)
 360                      (epsilon (sqrt machine-epsilon))
 361                      (count-limit 500)
 362                      (verbose t)
 363                      (alpha 1.0)
 364                      (beta 0.5)
 365                      (gamma 2.0)
 366                      (delta 0.5))
 367 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 368           (count-limit 500) (verbose t) alpha beta gamma delta)
 369 Maximizes F using the Nelder-Mead simplex method. START can be a
 370 starting simplex - a list of N+1 points, with N=dimension of problem,
 371 or a single point. If start is a single point you should give the
 372 size of the initial simplex as SIZE, a sequence of length N. Default is
 373 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 374 control the behavior of simplex algorithm."
 375     (let ((s (send simplex-proto :new f start size)))
 376       (do ((best (send s :best-point) (send s :best-point))
 377            (count 0 (+ count 1))
 378            next)
 379           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 380            (if (and verbose (>= count count-limit))
 381                (format t "Iteration limit exceeded.~%"))
 382            (send s :point-location (send s :best-point)))
 383           (setf next (send s :extrapolate-from-worst (- alpha)))
 384           (if (send s :is-worse best next)
 385               (setf next (send s :extrapolate-from-worst gamma))
 386               (when (send s :is-worse next (send s :second-worst-point))
 387                     (setf next (send s :extrapolate-from-worst beta))
 388                     (if (send s :is-worse next (send s :worst-point))
 389                         (send s :shrink-to-best delta))))
 390           (if verbose
 391               (format t "Value = ~10g~%"
 392                       (send s :point-value (send s :best-point)))))))
 393
 394
 395
 396 ;;;
 397 ;;; Simplex Points
 398 ;;;
 399
 400 (defmeth simplex-proto :make-point (x)
 401   (let ((f (send self :f)))
 402     (if f
 403         (let ((val (funcall f x)))
 404           (cons (if (consp val) (car val) val) x))
 405         (cons nil x))))
 406
 407 (defmeth simplex-proto :point-value (x) (car x))
 408
 409 (defmeth simplex-proto :point-location (x) (cdr x))
 410
 411 (defmeth simplex-proto :is-worse (x y)
 412   (< (send self :point-value x) (send self :point-value y)))
 413
 414 ;;;
 415 ;;; Making New Simplices
 416 ;;;
 417
 418 (defmeth simplex-proto :isnew (f start &optional size)
 419   (send self :simplex start size)
 420   (send self :f f))
 421
 422 ;;;
 423 ;;; Slot Accessors and Mutators
 424 ;;;
 425
 426 (defmeth simplex-proto :simplex (&optional new size)
 427   (if new
 428       (let ((simplex
 429              (if (and (consp new) (sequencep (car new)))
 430                  (if (/= (length new) (+ 1 (length (car new))))
 431                      (error "bad simplex data")
 432                      (copy-list new))
 433                  (let* ((n (length new))
 434                         (size (if size size (repeat 1 n)))
 435                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 436                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 437                         (pts (cons (repeat 0 n)
 438                                    (mapcar #'(lambda (x) (coerce x 'list))
 439                                            (column-list (diagonal diag))))))
 440                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 441         (setf (slot-value 'simplex)
 442               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 443         (send self :sort-simplex)))
 444   (slot-value 'simplex))
 445
 446 (defmeth simplex-proto :f (&optional f)
 447   (when f
 448         (setf (slot-value 'f) f)
 449         (let ((simplex
 450                (mapcar #'(lambda (x) (send self :point-location x))
 451                        (send self :simplex))))
 452           (send self :simplex simplex)))
 453   (slot-value 'f))
 454
 455 (defmeth simplex-proto :sort-simplex ()
 456   (if (send self :f)
 457       (setf (slot-value 'simplex)
 458             (sort (slot-value 'simplex)
 459                   #'(lambda (x y) (send self :is-worse x y))))))
 460
 461 ;;;
 462 ;;; Other Methods Using List Representation of SImplex
 463 ;;;
 464
 465 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 466 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 467 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 468 (defmeth simplex-proto :replace-point (new old)
 469   (let* ((simplex (send self :simplex))
 470          (n (position old simplex)))
 471     (when n
 472           (setf (nth n simplex) new)
 473           (send self :sort-simplex))))
 474 (defmeth simplex-proto :mean-opposite-face (x)
 475   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 476                       (remove x (send self :simplex)))))
 477     (/ (reduce #'+ face) (length face))))
 478
 479 ;;;
 480 ;;; Iteration Step Methods
 481 ;;;
 482
 483 (defmeth simplex-proto :extrapolate-from-worst (fac)
 484   (let* ((worst (send self :worst-point))
 485          (wloc (send self :point-location worst))
 486          (delta (- (send self :mean-opposite-face worst) wloc))
 487          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 488     (if (send self :is-worse worst new) (send self :replace-point new worst))
 489     new))
 490
 491 (defmeth simplex-proto :shrink-to-best (fac)
 492   (let* ((best (send self :best-point))
 493          (bloc (send self :point-location best)))
 494     (dolist (x (copy-list (send self :simplex)))
 495             (if (not (eq x best))
 496                 (send self :replace-point
 497                       (send self :make-point
 498                             (+ bloc
 499                                (* fac
 500                                   (- (send self :point-location x) bloc))))
 501                       x)))))
 502
 503 (defmeth simplex-proto :relative-range ()
 504   (let ((best (send self :point-value (send self :best-point)))
 505         (worst (send self :point-value (send self :worst-point))))
 506     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 507
 508
 509
 510
 511 ;;;;
 512 ;;;; Maximization and Numerical Derivatives
 513 ;;;;
 514
 515
 516 (defun data2double (n data ptr)
 517   (declare (fixnum n))
 518   (let* ((seq (compound-data-seq data))
 519          (elem (make-next-element seq)))
 520     (if (/= (length seq) n) (error "bad data size"))
 521     (dotimes (i n)
 522       (declare (fixnum i))
 523       (la-put-double ptr i (get-next-element elem i)))))
 524
 525 (defun maximize-callback (n px pfval pgrad phess pderivs)
 526   (la-vector-to-data px n +mode-re+ *maximize-callback-arg*)
 527   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 528          (derivs (if (consp val) (- (length val) 1) 0)))
 529     (la-put-integer pderivs 0 derivs)
 530     (la-put-double pfval 0 (if (consp val) (first val) val))
 531     (if (<= 1 derivs) (data2double n (second val) pgrad))
 532     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 533
 534 (defun numgrad (f x &optional scale (h -1.0))
 535 "Args: (f x &optional scale derivstep)
 536 Computes the numerical gradient of F at X."
 537   (check-sequence x)
 538   (check-real x)
 539   (when scale
 540         (check-sequence scale)
 541         (check-real scale))
 542   (check-one-real h)
 543   (let* ((n (length x))
 544          (result (make-list n)))
 545     (if (and scale (/= n (length scale)))
 546         (error "scale not the same length as x"))
 547     (let ((*maximize-callback-function* f)
 548           (*maximize-callback-arg* (make-list n)))
 549       (let ((px (la-data-to-vector x +mode-re+))
 550             (pgrad (la-vector n +mode-re+))
 551             (pscale (la-data-to-vector
 552                      (if scale scale (make-list n :initial-element 1.0))
 553                      +mode-re+)))
 554         (unwind-protect
 555             (progn
 556               (numgrad-front n px pgrad h pscale)
 557               (la-vector-to-data pgrad n +mode-re+ result))
 558           (la-free-vector px)
 559           (la-free-vector pgrad)
 560           (la-free-vector pscale))))
 561       result))
 562
 563 (defun numhess (f x &optional scale (h -1.0) all)
 564 "Args: (f x &optional scale derivstep)
 565 Computes the numerical Hessian matrix of F at X."
 566   (check-sequence x)
 567   (check-real x)
 568   (when scale
 569         (check-sequence scale)
 570         (check-real scale))
 571   (check-one-real h)
 572   (let* ((n (length x))
 573          (result (if all
 574                      (list nil (make-list n) (make-array (list n n)))
 575                      (make-array (list n n)))))
 576     (if (and scale (/= n (length scale)))
 577         (error "scale not the same length as x"))
 578     (let ((*maximize-callback-function* f)
 579           (*maximize-callback-arg* (make-list n)))
 580       (let ((hess-data (compound-data-seq (if all (third result) result)))
 581             (px (la-data-to-vector x +mode-re+))
 582             (pf (la-vector 1 +mode-re+))
 583             (pgrad (la-vector n +mode-re+))
 584             (phess (la-vector (* n n) +mode-re+))
 585             (pscale (la-data-to-vector
 586                      (if scale scale (make-list n :initial-element 1.0))
 587                      +mode-re+)))
 588         (unwind-protect
 589             (progn
 590               (numhess-front n px pf pgrad phess h pscale)
 591               (when all
 592                     (setf (first result) (la-get-double pf 0))
 593                     (la-vector-to-data pgrad n +mode-re+ (second result)))
 594               (la-vector-to-data phess (* n n) +mode-re+ hess-data))
 595           (la-free-vector pf)
 596           (la-free-vector px)
 597           (la-free-vector pgrad)
 598           (la-free-vector phess)
 599           (la-free-vector pscale))))
 600     result))
 601