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 (defproto minfo-proto '(internals))
 198
 199 #+xlisp (send minfo-proto :add-method :isnew #'|minfo-isnew|)
 200 #+xlisp (send minfo-proto :add-method :maximize #'|minfo-maximize|)
 201 #+xlisp (send minfo-proto :add-method :loglaplace #'|minfo-loglap|)
 202 #-xlisp
 203 (defmeth minfo-proto :isnew (&rest args)
 204   (setf (slot-value 'internals) (apply #'new-minfo-internals args)))
 205 #-xlisp
 206 (defmeth minfo-proto :maximize (&rest args)
 207   (apply #'minfo-maximize (slot-value 'internals) args))
 208
 209 (defmeth minfo-proto :x () (aref (slot-value 'internals) 3))
 210 (defmeth minfo-proto :scale () (aref (slot-value 'internals) 4))
 211 (defmeth minfo-proto :derivstep () (aref (aref (slot-value 'internals) 9) 1))
 212 (defmeth minfo-proto :tilt () (aref (aref (slot-value 'internals) 9) 6))
 213
 214 (defmeth minfo-proto :f (&optional (val nil set))
 215   (when set
 216         (send self :set-no-vals-supplied)
 217         (setf (aref (slot-value 'internals) 0) val))
 218   (aref (slot-value 'internals) 0))
 219
 220 (defmeth minfo-proto :set-no-vals-supplied ()
 221   (setf (aref (aref (slot-value 'internals) 8) 6) 0))
 222
 223 (defmeth minfo-proto :exptilt (&optional (val nil set))
 224   (if set
 225       (let ((old (send self :exptilt)))
 226         (setf (aref (aref (slot-value 'internals) 8) 7) (if val 1 0))
 227         (if (and (not (or (and old val) (and (not old) (not val))))
 228                  (/= (send self :tilt) 0.0))
 229             (send self :set-no-vals-supplied))))
 230   (= 1 (aref (aref (slot-value 'internals) 8) 7)))
 231
 232 (defmeth minfo-proto :newtilt (&optional (val nil set))
 233   (when set
 234         (setf (aref (aref (slot-value 'internals) 9) 7) (float val))
 235         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 236   (aref (aref (slot-value 'internals) 9) 7))
 237
 238 (defmeth minfo-proto :gfuns (&optional (val nil set))
 239   (when set
 240         (if (or (not (consp val))
 241                 (not (every #'functionp val)))
 242             (error "not all functions"))
 243         (setf (aref (slot-value 'internals) 1) val)
 244         (setf (aref (aref (slot-value 'internals) 8) 1) (length val))
 245         (setf (aref (slot-value 'internals) 10) (repeat 1.0 (length val)))
 246         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 247   (aref (slot-value 'internals) 1))
 248
 249 (defmeth minfo-proto :cfuns (&optional (val nil set))
 250   (when set
 251         (if (or (not (consp val))
 252                 (not (every #'functionp val)))
 253             (error "not all functions"))
 254         (setf (aref (slot-value 'internals) 2) val)
 255         (setf (aref (aref (slot-value 'internals) 8) 2) (length val))
 256         (setf (aref (slot-value 'internals) 7) (repeat 0.0 (length val)))
 257         (setf (aref (slot-value 'internals) 11) (repeat 0.0 (length val)))
 258         (send self :set-no-vals-supplied))
 259   (aref (slot-value 'internals) 2))
 260
 261 (defmeth minfo-proto :ctarget (&optional (val nil set))
 262   (when set
 263         (if (/= (length val) (length (send self :ctarget)))
 264             (error "bad target length"))
 265         (setf (aref (slot-value 'internals) 7) val))
 266   (aref (slot-value 'internals) 7))
 267
 268 (defmeth minfo-proto :fvals ()
 269   (let* ((fv (aref (slot-value 'internals) 5))
 270          (n (length (send self :x)))
 271          (val (select fv 0))
 272          (grad (select fv (iseq 1 n)))
 273          (hess (matrix (list n n) (select fv (iseq (+ 1 n) (+ n (* n n)))))))
 274     (list val grad hess)))
 275
 276 (defmeth minfo-proto :copy ()
 277   (let ((obj (make-object minfo-proto))
 278         (internals (copy-seq (slot-value 'internals))))
 279     (dotimes (i (length internals))
 280              (let ((x (aref internals i)))
 281                (if (sequencep x)
 282                    (setf (aref internals i) (copy-seq x)))))
 283     (send obj :add-slot 'internals internals)
 284     obj))
 285
 286 (defmeth minfo-proto :derivscale ()
 287   (let* ((step (^ machine-epsilon (/ 1 6)))
 288          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 289          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 290     (setf hess (numhess (send self :f) (send self :x) scale step))
 291     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 292     (setf (aref (slot-value 'internals) 4) scale)
 293     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 294
 295 (defmeth minfo-proto :verbose (&optional (val nil set))
 296   (when set
 297         (setf (aref (aref (slot-value 'internals) 8) 5)
 298               (cond ((integerp val) val)
 299                     ((null val) 0)
 300                     (t 1))))
 301   (aref (aref (slot-value 'internals) 8) 5))
 302
 303 (defmeth minfo-proto :backtrack (&optional (val nil set))
 304   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 305   (aref (aref (slot-value 'internals) 8) 4))
 306
 307 (defmeth minfo-proto :maxiter (&optional (val nil set))
 308     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 309                   (if (integerp val) val -1)))
 310     (aref (aref (slot-value 'internals) 8) 3))
 311
 312 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 313   (when set
 314         (if (/= (length val) (length (send self :gfuns)))
 315             (error "wrong size tilt scale sequence"))
 316         (setf (aref (slot-value 'internals) 10) val))
 317   (aref (slot-value 'internals) 10))
 318
 319 ;;;;
 320 ;;;;
 321 ;;;; Newton's Method with Backtracking
 322 ;;;;
 323 ;;;;
 324
 325 (defun newtonmax (f start &key
 326                     scale
 327                     (derivstep -1.0)
 328                     (count-limit -1)
 329                     (verbose 1)
 330                     return-derivs)
 331 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 332 Maximizes F starting from START using Newton's method with backtracking.
 333 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 334 list of location, unction value, gradient and hessian at maximum.
 335 SCALE should be a list of the typical magnitudes of the parameters.
 336 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 337 of iteration information. COUNT-LIMIT limits the number of iterations"
 338   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 339         (minfo (send minfo-proto :new f start
 340                      :scale scale :derivstep derivstep)))
 341     (send minfo :maxiter count-limit)
 342     (send minfo :derivscale)
 343     (send minfo :maximize verbose)
 344     (if return-derivs
 345         (cons (send minfo :x) (- (send minfo :fvals)))
 346         (send minfo :x))))
 347
 348 ;;;;
 349 ;;;;
 350 ;;;; Nelder-Mead Simplex Method
 351 ;;;;
 352 ;;;;
 353
 354 (defun nelmeadmax (f start &key
 355                      (size 1)
 356                      (epsilon (sqrt machine-epsilon))
 357                      (count-limit 500)
 358                      (verbose t)
 359                      (alpha 1.0)
 360                      (beta 0.5)
 361                      (gamma 2.0)
 362                      (delta 0.5))
 363 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 364           (count-limit 500) (verbose t) alpha beta gamma delta)
 365 Maximizes F using the Nelder-Mead simplex method. START can be a
 366 starting simplex - a list of N+1 points, with N=dimension of problem,
 367 or a single point. If start is a single point you should give the
 368 size of the initial simplex as SIZE, a sequence of length N. Default is
 369 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 370 control the behavior of simplex algorithm."
 371     (let ((s (send simplex-proto :new f start size)))
 372       (do ((best (send s :best-point) (send s :best-point))
 373            (count 0 (+ count 1))
 374            next)
 375           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 376            (if (and verbose (>= count count-limit))
 377                (format t "Iteration limit exceeded.~%"))
 378            (send s :point-location (send s :best-point)))
 379           (setf next (send s :extrapolate-from-worst (- alpha)))
 380           (if (send s :is-worse best next)
 381               (setf next (send s :extrapolate-from-worst gamma))
 382               (when (send s :is-worse next (send s :second-worst-point))
 383                     (setf next (send s :extrapolate-from-worst beta))
 384                     (if (send s :is-worse next (send s :worst-point))
 385                         (send s :shrink-to-best delta))))
 386           (if verbose
 387               (format t "Value = ~10g~%"
 388                       (send s :point-value (send s :best-point)))))))
 389
 390
 391 ;;;
 392 ;;; Simplex Prototype
 393 ;;;
 394
 395 (defproto simplex-proto '(f simplex))
 396
 397 ;;;
 398 ;;; Simplex Points
 399 ;;;
 400
 401 (defmeth simplex-proto :make-point (x)
 402   (let ((f (send self :f)))
 403     (if f
 404         (let ((val (funcall f x)))
 405           (cons (if (consp val) (car val) val) x))
 406         (cons nil x))))
 407
 408 (defmeth simplex-proto :point-value (x) (car x))
 409
 410 (defmeth simplex-proto :point-location (x) (cdr x))
 411
 412 (defmeth simplex-proto :is-worse (x y)
 413   (< (send self :point-value x) (send self :point-value y)))
 414
 415 ;;;
 416 ;;; Making New Simplices
 417 ;;;
 418
 419 (defmeth simplex-proto :isnew (f start &optional size)
 420   (send self :simplex start size)
 421   (send self :f f))
 422
 423 ;;;
 424 ;;; Slot Accessors and Mutators
 425 ;;;
 426
 427 (defmeth simplex-proto :simplex (&optional new size)
 428   (if new
 429       (let ((simplex
 430              (if (and (consp new) (sequencep (car new)))
 431                  (if (/= (length new) (+ 1 (length (car new))))
 432                      (error "bad simplex data")
 433                      (copy-list new))
 434                  (let* ((n (length new))
 435                         (size (if size size (repeat 1 n)))
 436                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 437                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 438                         (pts (cons (repeat 0 n)
 439                                    (mapcar #'(lambda (x) (coerce x 'list))
 440                                            (column-list (diagonal diag))))))
 441                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 442         (setf (slot-value 'simplex)
 443               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 444         (send self :sort-simplex)))
 445   (slot-value 'simplex))
 446
 447 (defmeth simplex-proto :f (&optional f)
 448   (when f
 449         (setf (slot-value 'f) f)
 450         (let ((simplex
 451                (mapcar #'(lambda (x) (send self :point-location x))
 452                        (send self :simplex))))
 453           (send self :simplex simplex)))
 454   (slot-value 'f))
 455
 456 (defmeth simplex-proto :sort-simplex ()
 457   (if (send self :f)
 458       (setf (slot-value 'simplex)
 459             (sort (slot-value 'simplex)
 460                   #'(lambda (x y) (send self :is-worse x y))))))
 461
 462 ;;;
 463 ;;; Other Methods Using List Representation of SImplex
 464 ;;;
 465
 466 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 467 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 468 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 469 (defmeth simplex-proto :replace-point (new old)
 470   (let* ((simplex (send self :simplex))
 471          (n (position old simplex)))
 472     (when n
 473           (setf (nth n simplex) new)
 474           (send self :sort-simplex))))
 475 (defmeth simplex-proto :mean-opposite-face (x)
 476   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 477                       (remove x (send self :simplex)))))
 478     (/ (reduce #'+ face) (length face))))
 479
 480 ;;;
 481 ;;; Iteration Step Methods
 482 ;;;
 483
 484 (defmeth simplex-proto :extrapolate-from-worst (fac)
 485   (let* ((worst (send self :worst-point))
 486          (wloc (send self :point-location worst))
 487          (delta (- (send self :mean-opposite-face worst) wloc))
 488          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 489     (if (send self :is-worse worst new) (send self :replace-point new worst))
 490     new))
 491
 492 (defmeth simplex-proto :shrink-to-best (fac)
 493   (let* ((best (send self :best-point))
 494          (bloc (send self :point-location best)))
 495     (dolist (x (copy-list (send self :simplex)))
 496             (if (not (eq x best))
 497                 (send self :replace-point
 498                       (send self :make-point
 499                             (+ bloc
 500                                (* fac
 501                                   (- (send self :point-location x) bloc))))
 502                       x)))))
 503
 504 (defmeth simplex-proto :relative-range ()
 505   (let ((best (send self :point-value (send self :best-point)))
 506         (worst (send self :point-value (send self :worst-point))))
 507     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 508
 509
 510
 511
 512 ;;;;
 513 ;;;; Maximization and Numerical Derivatives
 514 ;;;;
 515
 516
 517 (defun data2double (n data ptr)
 518   (declare (fixnum n))
 519   (let* ((seq (compound-data-seq data))
 520          (elem (make-next-element seq)))
 521     (if (/= (length seq) n) (error "bad data size"))
 522     (dotimes (i n)
 523       (declare (fixnum i))
 524       (la-put-double ptr i (get-next-element elem i)))))
 525
 526 (defun maximize-callback (n px pfval pgrad phess pderivs)
 527   (la-vector-to-data px n +mode-re+ *maximize-callback-arg*)
 528   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 529          (derivs (if (consp val) (- (length val) 1) 0)))
 530     (la-put-integer pderivs 0 derivs)
 531     (la-put-double pfval 0 (if (consp val) (first val) val))
 532     (if (<= 1 derivs) (data2double n (second val) pgrad))
 533     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 534
 535 (defun numgrad (f x &optional scale (h -1.0))
 536 "Args: (f x &optional scale derivstep)
 537 Computes the numerical gradient of F at X."
 538   (check-sequence x)
 539   (check-real x)
 540   (when scale
 541         (check-sequence scale)
 542         (check-real scale))
 543   (check-one-real h)
 544   (let* ((n (length x))
 545          (result (make-list n)))
 546     (if (and scale (/= n (length scale)))
 547         (error "scale not the same length as x"))
 548     (let ((*maximize-callback-function* f)
 549           (*maximize-callback-arg* (make-list n)))
 550       (let ((px (la-data-to-vector x +mode-re+))
 551             (pgrad (la-vector n +mode-re+))
 552             (pscale (la-data-to-vector
 553                      (if scale scale (make-list n :initial-element 1.0))
 554                      +mode-re+)))
 555         (unwind-protect
 556             (progn
 557               (numgrad-front n px pgrad h pscale)
 558               (la-vector-to-data pgrad n +mode-re+ result))
 559           (la-free-vector px)
 560           (la-free-vector pgrad)
 561           (la-free-vector pscale))))
 562       result))
 563
 564 (defun numhess (f x &optional scale (h -1.0) all)
 565 "Args: (f x &optional scale derivstep)
 566 Computes the numerical Hessian matrix of F at X."
 567   (check-sequence x)
 568   (check-real x)
 569   (when scale
 570         (check-sequence scale)
 571         (check-real scale))
 572   (check-one-real h)
 573   (let* ((n (length x))
 574          (result (if all
 575                      (list nil (make-list n) (make-array (list n n)))
 576                      (make-array (list n n)))))
 577     (if (and scale (/= n (length scale)))
 578         (error "scale not the same length as x"))
 579     (let ((*maximize-callback-function* f)
 580           (*maximize-callback-arg* (make-list n)))
 581       (let ((hess-data (compound-data-seq (if all (third result) result)))
 582             (px (la-data-to-vector x +mode-re+))
 583             (pf (la-vector 1 +mode-re+))
 584             (pgrad (la-vector n +mode-re+))
 585             (phess (la-vector (* n n) +mode-re+))
 586             (pscale (la-data-to-vector
 587                      (if scale scale (make-list n :initial-element 1.0))
 588                      +mode-re+)))
 589         (unwind-protect
 590             (progn
 591               (numhess-front n px pf pgrad phess h pscale)
 592               (when all
 593                     (setf (first result) (la-get-double pf 0))
 594                     (la-vector-to-data pgrad n +mode-re+ (second result)))
 595               (la-vector-to-data phess (* n n) +mode-re+ hess-data))
 596           (la-free-vector pf)
 597           (la-free-vector px)
 598           (la-free-vector pgrad)
 599           (la-free-vector phess)
 600           (la-free-vector pscale))))
 601     result))
 602