src/numerics/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        :cffi
  11        :lisp-stat-ffi-int
  12        :lisp-stat-object-system
  13        :lisp-stat-types
  14        :lisp-stat-compound-data
  15        :lisp-stat-math
  16        :lisp-stat-float
  17        :lisp-stat-basics
  18        :lisp-stat-matrix
  19        :lisp-stat-linalg-data
  20        :lisp-stat-linalg)
  21  (:shadowing-import-from :lisp-stat-object-system
  22                          slot-value call-method call-next-method)
  23  (:shadowing-import-from :lisp-stat-math
  24            expt + - * / ** mod rem abs 1+ 1- log exp sqrt sin cos tan
  25            asin acos atan sinh cosh tanh asinh acosh atanh float random
  26            truncate floor ceiling round minusp zerop plusp evenp oddp
  27            < <= = /= >= > complex conjugate realpart imagpart phase
  28            min max logand logior logxor lognot ffloor fceiling
  29            ftruncate fround signum cis)
  30  (:export
  31      ;; derivatives
  32      numgrad numhess
  33
  34      ;; optimization
  35      newtonmax nelmeadmax))
  36
  37 ;; matrix is in statistics, but should that be a predecessor?
  38
  39 ;;; FIXME:AJR: There is a need to figure out the proper symbols to
  40 ;;; export.  more importantly should there be any specialty package
  41 ;;; that are exported for maximization?
  42
  43 (in-package :lisp-stat-optimize)
  44
  45 #+openmcl
  46 (defctype size-t :unsigned-long)
  47 #+sbcl
  48 (defctype size-t :unsigned-int)
  49
  50 (defvar *maximize-callback-function* nil
  51   "Used in generic optimization to determine function name -- symbol or string?")
  52
  53 (defvar *maximize-callback-arg* nil
  54   "args to function to maximize")
  55
  56
  57 ;;;
  58 ;;; CFFI support using library for optimization work.
  59 ;;;
  60
  61 ;; There is a problem with this particular approach, in terms of
  62 ;; circular dependencies.  We can not have this out-of-object call
  63 ;; into optimize, at least not from here.
  64 (cffi:defcallback ccl-maximize-callback :void ((n :int)
  65                                                (px :pointer)
  66                                                (pfval :pointer)
  67                                                (pgrad :pointer)
  68                                                (phess :pointer)
  69                                                (pderivs :pointer))
  70    (lisp-stat-optimize::maximize-callback n px pfval pgrad phess pderivs))
  71
  72 (cffi:defcfun ("register_maximize_callback" register-maximize-callback)
  73     :void (x :pointer))
  74 (register-maximize-callback (cffi:callback ccl-maximize-callback))
  75
  76 (cffi:defcfun ("ccl_numgrad_front" ccl-numgrad-front)
  77     :void (x size-t) (y :pointer) (z :pointer) (u :double) (v :pointer))
  78 (defun numgrad-front (x y z u v)
  79   (ccl-numgrad-front x y z (float u 1d0) v))
  80
  81 (cffi:defcfun ("ccl_numhess_front" ccl-numhess-front)
  82     :void (x size-t) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :double) (a :pointer))
  83 (defun numhess-front (x y z u v w a)
  84   (ccl-numhess-front x y z u v (float w 1d0) a))
  85
  86 (cffi:defcfun ("ccl_minfo_maximize" ccl-minfo-maximize)
  87     :void (x :pointer) (y :pointer) (z :pointer) (u :pointer) (v :pointer) (w :int))
  88 (defun base-minfo-maximize (x y z u v w)
  89   (ccl-minfo-maximize x y z u v w))
  90
  91
  92
  93 ;;;;
  94 ;;;; minfo basics (internal??)
  95 ;;;;
  96
  97 (defun init-minfo-ipar-values (n ipars &key
  98                                (TRUE 1)
  99                                (FALSE 0)
 100                                (k 0)
 101                                (m 0)
 102                                (itnlimit -1)
 103                                (backtrack TRUE)
 104                                (verbose 0)
 105                                (vals_suppl FALSE)
 106                                (exptilt TRUE)
 107                                (count 0)
 108                                (termcode 0))
 109   "Initialize ipars (iteration parameters) by destructive modification."
 110   (setf (aref ipars 0) n)
 111   (setf (aref ipars 1) m)
 112   (setf (aref ipars 2) k)
 113   (setf (aref ipars 3) itnlimit)
 114   (setf (aref ipars 4) backtrack)
 115   (setf (aref ipars 5) verbose)
 116   (setf (aref ipars 6) vals_suppl)
 117   (setf (aref ipars 7) exptilt)
 118   (setf (aref ipars 8) count)
 119   (setf (aref ipars 9) termcode))
 120
 121 (defun init-minfo-dpar-values (h dpars &key
 122                                (typf 1.0)
 123                                (gradtol -1.0)
 124                                (steptol -1.0)
 125                                (maxstep -1.0)
 126                                (dflt 0.0)
 127                                (tilt 0.0)
 128                                (newtilt 0.0)
 129                                (hessadd 0.0))
 130   "Initialize dpars (derivative parameters) by destructive modification."
 131   (setf (aref dpars 0) typf)
 132   (setf (aref dpars 1) h)
 133   (setf (aref dpars 2) gradtol)
 134   (setf (aref dpars 3) steptol)
 135   (setf (aref dpars 4) maxstep)
 136   (setf (aref dpars 5) dflt)
 137   (setf (aref dpars 6) tilt)
 138   (setf (aref dpars 7) newtilt)
 139   (setf (aref dpars 8) hessadd))
 140
 141 (defun init-minfo-internals (n h internals)
 142   (let ((ipars (aref internals 8))
 143         (dpars (aref internals 9)))
 144     (init-minfo-ipar-values n ipars)
 145     (init-minfo-dpar-values h dpars)))
 146
 147 (defun new-minfo-internals (f x &key scale ((:derivstep h) -1.0))
 148   (check-sequence x)
 149   (check-real x)
 150   (check-one-real h)
 151   (let ((n (length x)))
 152     (when scale
 153           (check-sequence scale)
 154           (check-real scale)
 155           (if (/= n (length scale)) (error "scale and x not the same length")))
 156     (let ((internals (make-array 12)))
 157       (setf (aref internals 0) f)
 158       (setf (aref internals 3) (if (consp x) (copy-list x) (coerce x 'list)))
 159       (setf (aref internals 4)
 160             (if scale (copy-seq scale) (make-array n :initial-element 1.0)))
 161       (setf (aref internals 5) (make-list (+ 1 n (* n n))))
 162       (setf (aref internals 8) (make-array 10))
 163       (setf (aref internals 9) (make-array 9))
 164       (init-minfo-internals n h internals)
 165       internals)))
 166
 167 (defun minfo-maximize (internals &optional verbose)
 168   "This function does what?"
 169   (let* ((f (aref internals 0))
 170          (x (aref internals 3))
 171          (fvals (aref internals 5))
 172          (n (length x))
 173          (v (if verbose (if (integerp verbose) verbose 1) -1)))
 174     (setf (aref internals 3) (copy-list x))
 175     (setf (aref internals 5) (copy-list fvals))
 176     (let ((*maximize-callback-function* f)
 177           (*maximize-callback-arg* (make-list n)))
 178       (let* ((x (aref internals 3))
 179              (scale (aref internals 4))
 180              (fvals (aref internals 5))
 181              (ip (aref internals 8))
 182              (dp (aref internals 9))
 183              (px (la-data-to-vector x +mode-re+))
 184              (pscale (la-data-to-vector scale +mode-re+))
 185              (pfvals (la-vector (length fvals) +mode-re+))
 186              (pip (la-data-to-vector ip +mode-in+))
 187              (pdp (la-data-to-vector dp +mode-re+)))
 188         (unwind-protect
 189             (progn
 190               (base-minfo-maximize px pfvals pscale pip pdp v)) ;; access to C
 191           (la-vector-to-data px n +mode-re+ x)
 192           (la-vector-to-data pfvals (+ 1 n (* n n)) +mode-re+ fvals)
 193           (la-vector-to-data pip (length ip) +mode-in+ ip)
 194           (la-vector-to-data pdp (length dp) +mode-re+ dp))
 195         (get-buf)))))
 196
 197
 198
 199 ;;;;
 200 ;;;; Mode Info Prototype
 201 ;;;;
 202
 203 (defvar minfo-proto)
 204 (defproto minfo-proto '(internals))
 205
 206 #+xlisp (send minfo-proto :add-method :isnew #'|minfo-isnew|)
 207 #+xlisp (send minfo-proto :add-method :maximize #'|minfo-maximize|)
 208 #+xlisp (send minfo-proto :add-method :loglaplace #'|minfo-loglap|)
 209 #-xlisp
 210 (defmeth minfo-proto :isnew (&rest args)
 211   (setf (slot-value 'internals) (apply #'new-minfo-internals args)))
 212 #-xlisp
 213 (defmeth minfo-proto :maximize (&rest args)
 214   (apply #'minfo-maximize (slot-value 'internals) args))
 215
 216 (defmeth minfo-proto :x () (aref (slot-value 'internals) 3))
 217 (defmeth minfo-proto :scale () (aref (slot-value 'internals) 4))
 218 (defmeth minfo-proto :derivstep () (aref (aref (slot-value 'internals) 9) 1))
 219 (defmeth minfo-proto :tilt () (aref (aref (slot-value 'internals) 9) 6))
 220
 221 (defmeth minfo-proto :f (&optional (val nil set))
 222   (when set
 223         (send self :set-no-vals-supplied)
 224         (setf (aref (slot-value 'internals) 0) val))
 225   (aref (slot-value 'internals) 0))
 226
 227 (defmeth minfo-proto :set-no-vals-supplied ()
 228   (setf (aref (aref (slot-value 'internals) 8) 6) 0))
 229
 230 (defmeth minfo-proto :exptilt (&optional (val nil set))
 231   (if set
 232       (let ((old (send self :exptilt)))
 233         (setf (aref (aref (slot-value 'internals) 8) 7) (if val 1 0))
 234         (if (and (not (or (and old val) (and (not old) (not val))))
 235                  (/= (send self :tilt) 0.0))
 236             (send self :set-no-vals-supplied))))
 237   (= 1 (aref (aref (slot-value 'internals) 8) 7)))
 238
 239 (defmeth minfo-proto :newtilt (&optional (val nil set))
 240   (when set
 241         (setf (aref (aref (slot-value 'internals) 9) 7) (float val))
 242         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 243   (aref (aref (slot-value 'internals) 9) 7))
 244
 245 (defmeth minfo-proto :gfuns (&optional (val nil set))
 246   (when set
 247         (if (or (not (consp val))
 248                 (not (every #'functionp val)))
 249             (error "not all functions"))
 250         (setf (aref (slot-value 'internals) 1) val)
 251         (setf (aref (aref (slot-value 'internals) 8) 1) (length val))
 252         (setf (aref (slot-value 'internals) 10) (repeat 1.0 (length val)))
 253         (if (/= (send self :tilt) 0.0) (send self :set-no-vals-supplied)))
 254   (aref (slot-value 'internals) 1))
 255
 256 (defmeth minfo-proto :cfuns (&optional (val nil set))
 257   (when set
 258         (if (or (not (consp val))
 259                 (not (every #'functionp val)))
 260             (error "not all functions"))
 261         (setf (aref (slot-value 'internals) 2) val)
 262         (setf (aref (aref (slot-value 'internals) 8) 2) (length val))
 263         (setf (aref (slot-value 'internals) 7) (repeat 0.0 (length val)))
 264         (setf (aref (slot-value 'internals) 11) (repeat 0.0 (length val)))
 265         (send self :set-no-vals-supplied))
 266   (aref (slot-value 'internals) 2))
 267
 268 (defmeth minfo-proto :ctarget (&optional (val nil set))
 269   (when set
 270         (if (/= (length val) (length (send self :ctarget)))
 271             (error "bad target length"))
 272         (setf (aref (slot-value 'internals) 7) val))
 273   (aref (slot-value 'internals) 7))
 274
 275 (defmeth minfo-proto :fvals ()
 276   (let* ((fv (aref (slot-value 'internals) 5))
 277          (n (length (send self :x)))
 278          (val (select fv 0))
 279          (grad (select fv (iseq 1 n)))
 280          (hess (matrix (list n n) (select fv (iseq (+ 1 n) (+ n (* n n)))))))
 281     (list val grad hess)))
 282
 283 (defmeth minfo-proto :copy ()
 284 "Method: ()
 285
 286 Make a copy of an minfo instance."
 287   (let ((obj (make-object minfo-proto))
 288         (internals (copy-seq (slot-value 'internals))))
 289     (dotimes (i (length internals))
 290              (let ((x (aref internals i)))
 291                (if (typep x 'sequence)
 292                    (setf (aref internals i) (copy-seq x)))))
 293     (send obj :add-slot 'internals internals)
 294     obj))
 295
 296 (defmeth minfo-proto :derivscale ()
 297   (let* ((step (^ machine-epsilon (/ 1 6)))
 298          (hess (numhess (send self :f) (send self :x) (send self :scale) step))
 299          (scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess)))))))
 300     (setf hess (numhess (send self :f) (send self :x) scale step))
 301     (setf scale (pmax (abs (send self :x)) (sqrt (abs (/ (diagonal hess))))))
 302     (setf (aref (slot-value 'internals) 4) scale)
 303     (setf (aref (aref (slot-value 'internals) 9) 1) step)))
 304
 305 (defmeth minfo-proto :verbose (&optional (val nil set))
 306   (when set
 307         (setf (aref (aref (slot-value 'internals) 8) 5)
 308               (cond ((integerp val) val)
 309                     ((null val) 0)
 310                     (t 1))))
 311   (aref (aref (slot-value 'internals) 8) 5))
 312
 313 (defmeth minfo-proto :backtrack (&optional (val nil set))
 314   (if set (setf (aref (aref (slot-value 'internals) 8) 4) (if val 1 0)))
 315   (aref (aref (slot-value 'internals) 8) 4))
 316
 317 (defmeth minfo-proto :maxiter (&optional (val nil set))
 318     (if set (setf (aref (aref (slot-value 'internals) 8) 3)
 319                   (if (integerp val) val -1)))
 320     (aref (aref (slot-value 'internals) 8) 3))
 321
 322 (defmeth minfo-proto :tiltscale (&optional (val nil set))
 323   (when set
 324         (if (/= (length val) (length (send self :gfuns)))
 325             (error "wrong size tilt scale sequence"))
 326         (setf (aref (slot-value 'internals) 10) val))
 327   (aref (slot-value 'internals) 10))
 328
 329 ;;;;
 330 ;;;;
 331 ;;;; Newton's Method with Backtracking
 332 ;;;;
 333 ;;;;
 334
 335 (defun newtonmax (f start &key
 336                     scale
 337                     (derivstep -1.0)
 338                     (count-limit -1)
 339                     (verbose 1)
 340                     return-derivs)
 341 "Args:(f start &key scale derivstep (verbose 1) return-derivs)
 342 Maximizes F starting from START using Newton's method with backtracking.
 343 If RETURN-DERIVS is NIL returns location of maximum; otherwise returns
 344 list of location, unction value, gradient and hessian at maximum.
 345 SCALE should be a list of the typical magnitudes of the parameters.
 346 DERIVSTEP is used in numerical derivatives and VERBOSE controls printing
 347 of iteration information. COUNT-LIMIT limits the number of iterations"
 348   (let ((verbose (if verbose (if (integerp verbose) verbose 1) 0))
 349         (minfo (send minfo-proto :new f start
 350                      :scale scale :derivstep derivstep)))
 351     (send minfo :maxiter count-limit)
 352     (send minfo :derivscale)
 353     (send minfo :maximize verbose)
 354     (if return-derivs
 355         (cons (send minfo :x) (- (send minfo :fvals)))
 356         (send minfo :x))))
 357
 358 ;;;
 359 ;;; Nelder-Mead Simplex Method
 360 ;;;
 361
 362 ;;; Simplex Prototype
 363
 364 (defvar simplex-proto)
 365 (defproto simplex-proto '(f simplex))
 366
 367 (defun nelmeadmax (f start &key
 368                      (size 1)
 369                      (epsilon (sqrt machine-epsilon))
 370                      (count-limit 500)
 371                      (verbose t)
 372                      (alpha 1.0)
 373                      (beta 0.5)
 374                      (gamma 2.0)
 375                      (delta 0.5))
 376 "Args: (f start &key (size 1) (epsilon (sqrt machine-epsilon))
 377           (count-limit 500) (verbose t) alpha beta gamma delta)
 378 Maximizes F using the Nelder-Mead simplex method. START can be a
 379 starting simplex - a list of N+1 points, with N=dimension of problem,
 380 or a single point. If start is a single point you should give the
 381 size of the initial simplex as SIZE, a sequence of length N. Default is
 382 all 1's. EPSILON is the convergence tolerance. ALPHA-DELTA can be used to
 383 control the behavior of simplex algorithm."
 384     (let ((s (send simplex-proto :new f start size)))
 385       (do ((best (send s :best-point) (send s :best-point))
 386            (count 0 (+ count 1))
 387            next)
 388           ((or (< (send s :relative-range) epsilon) (>= count count-limit))
 389            (if (and verbose (>= count count-limit))
 390                (format t "Iteration limit exceeded.~%"))
 391            (send s :point-location (send s :best-point)))
 392           (setf next (send s :extrapolate-from-worst (- alpha)))
 393           (if (send s :is-worse best next)
 394               (setf next (send s :extrapolate-from-worst gamma))
 395               (when (send s :is-worse next (send s :second-worst-point))
 396                     (setf next (send s :extrapolate-from-worst beta))
 397                     (if (send s :is-worse next (send s :worst-point))
 398                         (send s :shrink-to-best delta))))
 399           (if verbose
 400               (format t "Value = ~10g~%"
 401                       (send s :point-value (send s :best-point)))))))
 402
 403
 404
 405 ;;;
 406 ;;; Simplex Points
 407 ;;;
 408
 409 (defmeth simplex-proto :make-point (x)
 410   (let ((f (send self :f)))
 411     (if f
 412         (let ((val (funcall f x)))
 413           (cons (if (consp val) (car val) val) x))
 414         (cons nil x))))
 415
 416 (defmeth simplex-proto :point-value (x) (car x))
 417
 418 (defmeth simplex-proto :point-location (x) (cdr x))
 419
 420 (defmeth simplex-proto :is-worse (x y)
 421   (< (send self :point-value x) (send self :point-value y)))
 422
 423 ;;;
 424 ;;; Making New Simplices
 425 ;;;
 426
 427 (defmeth simplex-proto :isnew (f start &optional size)
 428   (send self :simplex start size)
 429   (send self :f f))
 430
 431 ;;;
 432 ;;; Slot Accessors and Mutators
 433 ;;;
 434
 435 (defmeth simplex-proto :simplex (&optional new size)
 436   (if new
 437       (let ((simplex
 438              (if (and (consp new) (sequencep (car new)))
 439                  (if (/= (length new) (+ 1 (length (car new))))
 440                      (error "bad simplex data")
 441                      (copy-list new))
 442                  (let* ((n (length new))
 443                         (size (if size size (repeat 1 n)))
 444                      ;   (pts (- (* 2 (uniform-rand (repeat n (+ n 1)))) 1)))
 445                         (diag (* 2 size (- (random (repeat 2 n)) .5)))
 446                         (pts (cons (repeat 0 n)
 447                                    (mapcar #'(lambda (x) (coerce x 'list))
 448                                            (column-list (diagonal diag))))))
 449                    (mapcar #'(lambda (x) (reduce #'+ (list (* size x) new))) pts)))))
 450         (setf (slot-value 'simplex)
 451               (mapcar #'(lambda (x) (send self :make-point x)) simplex))
 452         (send self :sort-simplex)))
 453   (slot-value 'simplex))
 454
 455 (defmeth simplex-proto :f (&optional f)
 456   (when f
 457         (setf (slot-value 'f) f)
 458         (let ((simplex
 459                (mapcar #'(lambda (x) (send self :point-location x))
 460                        (send self :simplex))))
 461           (send self :simplex simplex)))
 462   (slot-value 'f))
 463
 464 (defmeth simplex-proto :sort-simplex ()
 465   (if (send self :f)
 466       (setf (slot-value 'simplex)
 467             (sort (slot-value 'simplex)
 468                   #'(lambda (x y) (send self :is-worse x y))))))
 469
 470 ;;;
 471 ;;; Other Methods Using List Representation of SImplex
 472 ;;;
 473
 474 (defmeth simplex-proto :best-point () (car (last (send self :simplex))))
 475 (defmeth simplex-proto :worst-point () (first (send self :simplex)))
 476 (defmeth simplex-proto :second-worst-point () (second (send self :simplex)))
 477 (defmeth simplex-proto :replace-point (new old)
 478   (let* ((simplex (send self :simplex))
 479          (n (position old simplex)))
 480     (when n
 481           (setf (nth n simplex) new)
 482           (send self :sort-simplex))))
 483 (defmeth simplex-proto :mean-opposite-face (x)
 484   (let ((face (mapcar #'(lambda (x) (send self :point-location x))
 485                       (remove x (send self :simplex)))))
 486     (/ (reduce #'+ face) (length face))))
 487
 488 ;;;
 489 ;;; Iteration Step Methods
 490 ;;;
 491
 492 (defmeth simplex-proto :extrapolate-from-worst (fac)
 493   (let* ((worst (send self :worst-point))
 494          (wloc (send self :point-location worst))
 495          (delta (- (send self :mean-opposite-face worst) wloc))
 496          (new (send self :make-point (+ wloc (* (- 1 fac) delta)))))
 497     (if (send self :is-worse worst new) (send self :replace-point new worst))
 498     new))
 499
 500 (defmeth simplex-proto :shrink-to-best (fac)
 501   (let* ((best (send self :best-point))
 502          (bloc (send self :point-location best)))
 503     (dolist (x (copy-list (send self :simplex)))
 504             (if (not (eq x best))
 505                 (send self :replace-point
 506                       (send self :make-point
 507                             (+ bloc
 508                                (* fac
 509                                   (- (send self :point-location x) bloc))))
 510                       x)))))
 511
 512 (defmeth simplex-proto :relative-range ()
 513   (let ((best (send self :point-value (send self :best-point)))
 514         (worst (send self :point-value (send self :worst-point))))
 515     (* 2 (/ (abs (- best worst)) (+ 1 (abs best) (abs worst))))))
 516
 517
 518
 519
 520 ;;;;
 521 ;;;; Maximization and Numerical Derivatives
 522 ;;;;
 523
 524
 525 (defun data2double (n data ptr)
 526   (declare (fixnum n))
 527   (let* ((seq (compound-data-seq data))
 528          (elem (make-next-element seq)))
 529     (if (/= (length seq) n) (error "bad data size"))
 530     (dotimes (i n)
 531       (declare (fixnum i))
 532       (la-put-double ptr i (get-next-element elem i)))))
 533
 534 (defun maximize-callback (n px pfval pgrad phess pderivs)
 535   (la-vector-to-data px n +mode-re+ *maximize-callback-arg*)
 536   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 537          (derivs (if (consp val) (- (length val) 1) 0)))
 538     (la-put-integer pderivs 0 derivs)
 539     (la-put-double pfval 0 (if (consp val) (first val) val))
 540     (if (<= 1 derivs) (data2double n (second val) pgrad))
 541     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 542
 543 (defun numgrad (f x &optional scale (h -1.0))
 544 "Args: (f x &optional scale derivstep)
 545 Computes the numerical gradient of F at X."
 546   (check-sequence x)
 547   (check-real x)
 548   (when scale
 549         (check-sequence scale)
 550         (check-real scale))
 551   (check-one-real h)
 552   (let* ((n (length x))
 553          (result (make-list n)))
 554     (if (and scale (/= n (length scale)))
 555         (error "scale not the same length as x"))
 556     (let ((*maximize-callback-function* f)
 557           (*maximize-callback-arg* (make-list n)))
 558       (let ((px (la-data-to-vector x +mode-re+))
 559             (pgrad (la-vector n +mode-re+))
 560             (pscale (la-data-to-vector
 561                      (if scale scale (make-list n :initial-element 1.0))
 562                      +mode-re+)))
 563         (unwind-protect
 564             (progn
 565               (numgrad-front n px pgrad h pscale)
 566               (la-vector-to-data pgrad n +mode-re+ result))
 567           (la-free-vector px)
 568           (la-free-vector pgrad)
 569           (la-free-vector pscale))))
 570       result))
 571
 572 (defun numhess (f x &optional scale (h -1.0) all)
 573 "Args: (f x &optional scale derivstep)
 574 Computes the numerical Hessian matrix of F at X."
 575   (check-sequence x)
 576   (check-real x)
 577   (when scale
 578         (check-sequence scale)
 579         (check-real scale))
 580   (check-one-real h)
 581   (let* ((n (length x))
 582          (result (if all
 583                      (list nil (make-list n) (make-array (list n n)))
 584                      (make-array (list n n)))))
 585     (if (and scale (/= n (length scale)))
 586         (error "scale not the same length as x"))
 587     (let ((*maximize-callback-function* f)
 588           (*maximize-callback-arg* (make-list n)))
 589       (let ((hess-data (compound-data-seq (if all (third result) result)))
 590             (px (la-data-to-vector x +mode-re+))
 591             (pf (la-vector 1 +mode-re+))
 592             (pgrad (la-vector n +mode-re+))
 593             (phess (la-vector (* n n) +mode-re+))
 594             (pscale (la-data-to-vector
 595                      (if scale scale (make-list n :initial-element 1.0))
 596                      +mode-re+)))
 597         (unwind-protect
 598             (progn
 599               (numhess-front n px pf pgrad phess h pscale)
 600               (when all
 601                     (setf (first result) (la-get-double pf 0))
 602                     (la-vector-to-data pgrad n +mode-re+ (second result)))
 603               (la-vector-to-data phess (* n n) +mode-re+ hess-data))
 604           (la-free-vector pf)
 605           (la-free-vector px)
 606           (la-free-vector pgrad)
 607           (la-free-vector phess)
 608           (la-free-vector pscale))))
 609     result))
 610