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 "Method: ()
 279
 280 Make a copy of an minfo instance."
 281   (let ((obj (make-object minfo-proto))
 282         (internals (copy-seq (slot-value 'internals))))
 283     (dotimes (i (length internals))
 284              (let ((x (aref internals i)))
 285                (if (sequencep x)
 286                    (setf (aref internals i) (copy-seq x)))))
 287     (send obj :add-slot 'internals internals)
 288     obj))
 289
 290 (defmeth minfo-proto :derivscale ()
 291   (let* ((step (^ machine-epsilon (/ 1 6)))
 292          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 293          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 294     (setf hess (numhess (send self :f) (send self :x) scale step))
 295     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 296     (setf (aref (slot-value 'internals) 4) scale)
 297     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 298
 299 (defmeth minfo-proto :verbose (&optional (val nil set))
 300   (when set
 301         (setf (aref (aref (slot-value 'internals) 8) 5)
 302               (cond ((integerp val) val)
 303                     ((null val) 0)
 304                     (t 1))))
 305   (aref (aref (slot-value 'internals) 8) 5))
 306
 307 (defmeth minfo-proto :backtrack (&optional (val nil set))
 308   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 309   (aref (aref (slot-value 'internals) 8) 4))
 310
 311 (defmeth minfo-proto :maxiter (&optional (val nil set))
 312     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 313                   (if (integerp val) val -1)))
 314     (aref (aref (slot-value 'internals) 8) 3))
 315
 316 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 317   (when set
 318         (if (/= (length val) (length (send self :gfuns)))
 319             (error "wrong size tilt scale sequence"))
 320         (setf (aref (slot-value 'internals) 10) val))
 321   (aref (slot-value 'internals) 10))
 322
 323 ;;;;
 324 ;;;;
 325 ;;;; Newton's Method with Backtracking
 326 ;;;;
 327 ;;;;
 328
 329 (defun newtonmax (f start &key
 330                     scale
 331                     (derivstep -1.0)
 332                     (count-limit -1)
 333                     (verbose 1)
 334                     return-derivs)
 335 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 336 Maximizes F starting from START using Newton's method with backtracking.
 337 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 338 list of location, unction value, gradient and hessian at maximum.
 339 SCALE should be a list of the typical magnitudes of the parameters.
 340 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 341 of iteration information. COUNT-LIMIT limits the number of iterations"
 342   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 343         (minfo (send minfo-proto :new f start
 344                      :scale scale :derivstep derivstep)))
 345     (send minfo :maxiter count-limit)
 346     (send minfo :derivscale)
 347     (send minfo :maximize verbose)
 348     (if return-derivs
 349         (cons (send minfo :x) (- (send minfo :fvals)))
 350         (send minfo :x))))
 351
 352 ;;;
 353 ;;; Nelder-Mead Simplex Method
 354 ;;;
 355
 356 ;;; Simplex Prototype
 357
 358 (defvar simplex-proto)
 359 (defproto simplex-proto '(f simplex))
 360
 361 (defun nelmeadmax (f start &key
 362                      (size 1)
 363                      (epsilon (sqrt machine-epsilon))
 364                      (count-limit 500)
 365                      (verbose t)
 366                      (alpha 1.0)
 367                      (beta 0.5)
 368                      (gamma 2.0)
 369                      (delta 0.5))
 370 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 371           (count-limit 500) (verbose t) alpha beta gamma delta)
 372 Maximizes F using the Nelder-Mead simplex method. START can be a
 373 starting simplex - a list of N+1 points, with N=dimension of problem,
 374 or a single point. If start is a single point you should give the
 375 size of the initial simplex as SIZE, a sequence of length N. Default is
 376 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 377 control the behavior of simplex algorithm."
 378     (let ((s (send simplex-proto :new f start size)))
 379       (do ((best (send s :best-point) (send s :best-point))
 380            (count 0 (+ count 1))
 381            next)
 382           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 383            (if (and verbose (>= count count-limit))
 384                (format t "Iteration limit exceeded.~%"))
 385            (send s :point-location (send s :best-point)))
 386           (setf next (send s :extrapolate-from-worst (- alpha)))
 387           (if (send s :is-worse best next)
 388               (setf next (send s :extrapolate-from-worst gamma))
 389               (when (send s :is-worse next (send s :second-worst-point))
 390                     (setf next (send s :extrapolate-from-worst beta))
 391                     (if (send s :is-worse next (send s :worst-point))
 392                         (send s :shrink-to-best delta))))
 393           (if verbose
 394               (format t "Value = ~10g~%"
 395                       (send s :point-value (send s :best-point)))))))
 396
 397
 398
 399 ;;;
 400 ;;; Simplex Points
 401 ;;;
 402
 403 (defmeth simplex-proto :make-point (x)
 404   (let ((f (send self :f)))
 405     (if f
 406         (let ((val (funcall f x)))
 407           (cons (if (consp val) (car val) val) x))
 408         (cons nil x))))
 409
 410 (defmeth simplex-proto :point-value (x) (car x))
 411
 412 (defmeth simplex-proto :point-location (x) (cdr x))
 413
 414 (defmeth simplex-proto :is-worse (x y)
 415   (< (send self :point-value x) (send self :point-value y)))
 416
 417 ;;;
 418 ;;; Making New Simplices
 419 ;;;
 420
 421 (defmeth simplex-proto :isnew (f start &optional size)
 422   (send self :simplex start size)
 423   (send self :f f))
 424
 425 ;;;
 426 ;;; Slot Accessors and Mutators
 427 ;;;
 428
 429 (defmeth simplex-proto :simplex (&optional new size)
 430   (if new
 431       (let ((simplex
 432              (if (and (consp new) (sequencep (car new)))
 433                  (if (/= (length new) (+ 1 (length (car new))))
 434                      (error "bad simplex data")
 435                      (copy-list new))
 436                  (let* ((n (length new))
 437                         (size (if size size (repeat 1 n)))
 438                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 439                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 440                         (pts (cons (repeat 0 n)
 441                                    (mapcar #'(lambda (x) (coerce x 'list))
 442                                            (column-list (diagonal diag))))))
 443                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 444         (setf (slot-value 'simplex)
 445               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 446         (send self :sort-simplex)))
 447   (slot-value 'simplex))
 448
 449 (defmeth simplex-proto :f (&optional f)
 450   (when f
 451         (setf (slot-value 'f) f)
 452         (let ((simplex
 453                (mapcar #'(lambda (x) (send self :point-location x))
 454                        (send self :simplex))))
 455           (send self :simplex simplex)))
 456   (slot-value 'f))
 457
 458 (defmeth simplex-proto :sort-simplex ()
 459   (if (send self :f)
 460       (setf (slot-value 'simplex)
 461             (sort (slot-value 'simplex)
 462                   #'(lambda (x y) (send self :is-worse x y))))))
 463
 464 ;;;
 465 ;;; Other Methods Using List Representation of SImplex
 466 ;;;
 467
 468 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 469 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 470 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 471 (defmeth simplex-proto :replace-point (new old)
 472   (let* ((simplex (send self :simplex))
 473          (n (position old simplex)))
 474     (when n
 475           (setf (nth n simplex) new)
 476           (send self :sort-simplex))))
 477 (defmeth simplex-proto :mean-opposite-face (x)
 478   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 479                       (remove x (send self :simplex)))))
 480     (/ (reduce #'+ face) (length face))))
 481
 482 ;;;
 483 ;;; Iteration Step Methods
 484 ;;;
 485
 486 (defmeth simplex-proto :extrapolate-from-worst (fac)
 487   (let* ((worst (send self :worst-point))
 488          (wloc (send self :point-location worst))
 489          (delta (- (send self :mean-opposite-face worst) wloc))
 490          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 491     (if (send self :is-worse worst new) (send self :replace-point new worst))
 492     new))
 493
 494 (defmeth simplex-proto :shrink-to-best (fac)
 495   (let* ((best (send self :best-point))
 496          (bloc (send self :point-location best)))
 497     (dolist (x (copy-list (send self :simplex)))
 498             (if (not (eq x best))
 499                 (send self :replace-point
 500                       (send self :make-point
 501                             (+ bloc
 502                                (* fac
 503                                   (- (send self :point-location x) bloc))))
 504                       x)))))
 505
 506 (defmeth simplex-proto :relative-range ()
 507   (let ((best (send self :point-value (send self :best-point)))
 508         (worst (send self :point-value (send self :worst-point))))
 509     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 510
 511
 512
 513
 514 ;;;;
 515 ;;;; Maximization and Numerical Derivatives
 516 ;;;;
 517
 518
 519 (defun data2double (n data ptr)
 520   (declare (fixnum n))
 521   (let* ((seq (compound-data-seq data))
 522          (elem (make-next-element seq)))
 523     (if (/= (length seq) n) (error "bad data size"))
 524     (dotimes (i n)
 525       (declare (fixnum i))
 526       (la-put-double ptr i (get-next-element elem i)))))
 527
 528 (defun maximize-callback (n px pfval pgrad phess pderivs)
 529   (la-vector-to-data px n +mode-re+ *maximize-callback-arg*)
 530   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 531          (derivs (if (consp val) (- (length val) 1) 0)))
 532     (la-put-integer pderivs 0 derivs)
 533     (la-put-double pfval 0 (if (consp val) (first val) val))
 534     (if (<= 1 derivs) (data2double n (second val) pgrad))
 535     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 536
 537 (defun numgrad (f x &optional scale (h -1.0))
 538 "Args: (f x &optional scale derivstep)
 539 Computes the numerical gradient of F at X."
 540   (check-sequence x)
 541   (check-real x)
 542   (when scale
 543         (check-sequence scale)
 544         (check-real scale))
 545   (check-one-real h)
 546   (let* ((n (length x))
 547          (result (make-list n)))
 548     (if (and scale (/= n (length scale)))
 549         (error "scale not the same length as x"))
 550     (let ((*maximize-callback-function* f)
 551           (*maximize-callback-arg* (make-list n)))
 552       (let ((px (la-data-to-vector x +mode-re+))
 553             (pgrad (la-vector n +mode-re+))
 554             (pscale (la-data-to-vector
 555                      (if scale scale (make-list n :initial-element 1.0))
 556                      +mode-re+)))
 557         (unwind-protect
 558             (progn
 559               (numgrad-front n px pgrad h pscale)
 560               (la-vector-to-data pgrad n +mode-re+ result))
 561           (la-free-vector px)
 562           (la-free-vector pgrad)
 563           (la-free-vector pscale))))
 564       result))
 565
 566 (defun numhess (f x &optional scale (h -1.0) all)
 567 "Args: (f x &optional scale derivstep)
 568 Computes the numerical Hessian matrix of F at X."
 569   (check-sequence x)
 570   (check-real x)
 571   (when scale
 572         (check-sequence scale)
 573         (check-real scale))
 574   (check-one-real h)
 575   (let* ((n (length x))
 576          (result (if all
 577                      (list nil (make-list n) (make-array (list n n)))
 578                      (make-array (list n n)))))
 579     (if (and scale (/= n (length scale)))
 580         (error "scale not the same length as x"))
 581     (let ((*maximize-callback-function* f)
 582           (*maximize-callback-arg* (make-list n)))
 583       (let ((hess-data (compound-data-seq (if all (third result) result)))
 584             (px (la-data-to-vector x +mode-re+))
 585             (pf (la-vector 1 +mode-re+))
 586             (pgrad (la-vector n +mode-re+))
 587             (phess (la-vector (* n n) +mode-re+))
 588             (pscale (la-data-to-vector
 589                      (if scale scale (make-list n :initial-element 1.0))
 590                      +mode-re+)))
 591         (unwind-protect
 592             (progn
 593               (numhess-front n px pf pgrad phess h pscale)
 594               (when all
 595                     (setf (first result) (la-get-double pf 0))
 596                     (la-vector-to-data pgrad n +mode-re+ (second result)))
 597               (la-vector-to-data phess (* n n) +mode-re+ hess-data))
 598           (la-free-vector pf)
 599           (la-free-vector px)
 600           (la-free-vector pgrad)
 601           (la-free-vector phess)
 602           (la-free-vector pscale))))
 603     result))
 604