linalg.lsp

   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 ;;; what this should do:
   7 ;;; #1 - use CFFI (and possibly Verazanno) to import C/C++.
   8 ;;; #2 - what to do for Fortran?  Possibly: C <-> bridge, or CLapack?
   9 ;;;      problem: would be better to have access to Fortran.  For
  10 ;;;      example, think of Doug Bates comment on reverse-calls (as
  11 ;;;      distinct from callbacks).  It would be difficult if we don't
  12 ;;;      -- however, has anyone run Lapack or similar through F2CL?
  13 ;;;      Answer: yes, Matlisp does this.
  14
  15
  16 ;;;; linalg -- Lisp-Stat interface to basic linear algebra routines.
  17 ;;;;
  18 ;;;; Copyright (c) 1991, by Luke Tierney. Permission is granted for
  19 ;;;; unrestricted use.
  20
  21 ;;;;
  22 ;;;; Package Setup
  23 ;;;;
  24
  25 (in-package #:lisp-stat-basics)
  26
  27 (export '(chol-decomp lu-decomp lu-solve determinant inverse sv-decomp
  28           qr-decomp rcondest make-rotation spline kernel-dens kernel-smooth
  29           fft make-sweep-matrix sweep-operator ax+y numgrad numhess
  30           split-list eigen))
  31
  32 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  33 ;;;;
  34 ;;;;                 Lisp to C number conversion and checking
  35 ;;;;
  36 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  37
  38 ;;;;
  39 ;;;; Lisp to/from C sequence and matrix conversion and checking
  40 ;;;;
  41
  42 (defun is-cons (a)
  43   "FIXME:AJR this is not used anywhere?"
  44   (if (consp a) 1 0))
  45
  46 (defun check-fixnum (a)
  47   (if (/= 0 (la-data-mode a)) (error "not an integer sequence - ~s" a)))
  48
  49 (defun check-real (data)
  50   (let ((data (compound-data-seq data)))
  51     (cond
  52      ((vectorp data)
  53       (let ((n (length data)))
  54         (declare (fixnum n))
  55         (dotimes (i n)
  56                  (declare (fixnum i))
  57                  (check-one-real (aref data i)))))
  58      ((consp data) (dolist (x data) (check-one-real x)))
  59      (t (error "bad sequence - ~s" data)))))
  60
  61 (defun vec-assign (a i x) (setf (aref a i) x))
  62
  63 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  64 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  65 ;;;;
  66 ;;;;             Lisp Interfaces to Linear Algebra Routines
  67 ;;;;
  68 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  69 ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
  70
  71 ;;; FIXME: use dpbt[f2|rf], dpbstf, dpot[f2|rf]; dpptrf, zpbstf, zpbt[f2|rf]
  72 ;;; remember: factorization = decomposition, depending on training.
  73
  74 (defun chol-decomp (a &optional (maxoffl 0.0))
  75 "Args: (a)
  76 Modified Cholesky decomposition. A should be a square, symmetric matrix.
  77 Computes lower triangular matrix L such that L L^T = A + D where D is a
  78 diagonal matrix. If A is strictly positive definite D will be zero.
  79 Otherwise, D is as small as possible to make A + D numerically strictly
  80 positive definite. Returns a list (L (max D))."
  81   (check-square-matrix a)
  82   (check-real a)
  83   (let* ((n (array-dimension a 0))
  84          (result (make-array (list n n)))
  85          (dpars (list maxoffl 0.0)))
  86     (check-real dpars)
  87     (let ((mat (la-data-to-matrix a mode-re))
  88           (dp (la-data-to-vector dpars mode-re)))
  89       (unwind-protect
  90           (progn
  91             (chol-decomp-front mat n dp)
  92             (la-matrix-to-data mat n n mode-re result)
  93             (la-vector-to-data dp 2 mode-re dpars))
  94         (la-free-matrix mat n)
  95         (la-free-vector dp)))
  96     (list result (second dpars))))
  97
  98
  99 ;;; REPLACE with
 100 ;;;         (matlisp:lu M)
 101 ;;; i.e. result use by:
 102 ;;;         (setf (values (lu-out1 lu-out2 lu-out3)) (matlisp:lu my-matrix))
 103 ;;; for solution, ...
 104 ;;; for lu-solve:
 105 ;;;         (matlisp:gesv a b &opt ipivot)
 106
 107 (defun lu-decomp (a)
 108 "Args: (a)
 109 A is a square matrix of numbers (real or complex). Computes the LU
 110 decomposition of A and returns a list of the form (LU IV D FLAG), where
 111 LU is a matrix with the L part in the lower triangle, the U part in the
 112 upper triangle (the diagonal entries of L are taken to be 1), IV is a vector
 113 describing the row permutation used, D is 1 if the number of permutations
 114 is odd, -1 if even, and FLAG is T if A is numerically singular, NIL otherwise.
 115 Used bu LU-SOLVE."
 116   (check-square-matrix a)
 117   (let* ((n (array-dimension a 0))
 118          (mode (max mode-re (la-data-mode a)))
 119          (result (list (make-array (list n n)) (make-array n) nil nil)))
 120     (let ((mat (la-data-to-matrix a mode))
 121           (iv (la-vector n mode-in))
 122           (d (la-vector 1 mode-re))
 123           (singular 0))
 124       (unwind-protect
 125           (progn
 126             (setf singular (lu-decomp-front mat n iv mode d))
 127             (la-matrix-to-data mat n n mode (first result))
 128             (la-vector-to-data iv n mode-in (second result))
 129             (setf (third result) (la-get-double d 0))
 130             (setf (fourth result) (if (= singular 0.0) nil t)))
 131         (la-free-matrix mat n)
 132         (la-free-vector iv)
 133         (la-free-vector d)))
 134     result))
 135
 136 (defun lu-solve (lu lb)
 137 "Args: (lu b)
 138 LU is the result of (LU-DECOMP A) for a square matrix A, B is a sequence.
 139 Returns the solution to the equation Ax = B. Signals an error if A is
 140 singular."
 141   (let ((la (first lu))
 142         (lidx (second lu)))
 143     (check-square-matrix la)
 144     (check-sequence lidx)
 145     (check-sequence lb)
 146     (check-fixnum lidx)
 147     (let* ((n (num-rows la))
 148            (result (make-sequence (if (consp lb) 'list 'vector) n))
 149            (a-mode (la-data-mode la))
 150            (b-mode (la-data-mode lb)))
 151       (if (/= n (length lidx)) (error "index sequence is wrong length"))
 152       (if (/= n (length lb)) (error "right hand side is wrong length"))
 153       (let* ((mode (max mode-re a-mode b-mode))
 154              (a (la-data-to-matrix la mode))
 155              (indx (la-data-to-vector lidx mode-in))
 156              (b (la-data-to-vector lb mode))
 157              (singular 0))
 158         (unwind-protect
 159             (progn
 160               (setf singular (lu-solve-front a n indx b mode))
 161               (la-vector-to-data b n mode result))
 162           (la-free-matrix a n)
 163           (la-free-vector indx)
 164           (la-free-vector b))
 165         (if (/= 0.0 singular) (error "matrix is (numerically) singular"))
 166         result))))
 167
 168 (defun determinant (a)
 169 "Args: (m)
 170 Returns the determinant of the square matrix M."
 171   (let* ((lu (lu-decomp a))
 172          (la (first lu))
 173          (n (num-rows a))
 174          (d1 (third lu))
 175          (d2 0.d0))
 176     (declare (fixnum n))
 177     (flet ((fabs (x) (float (abs x) 0.d0)))
 178       (dotimes (i n (* d1 (exp d2)))
 179         (declare (fixnum i))
 180         (let* ((x (aref la i i))
 181                (magn (fabs x)))
 182           (if (= 0.0 magn) (return 0.d0))
 183           (setf d1 (* d1 (/ x magn)))
 184           (setf d2 (+ d2 (log magn))))))))
 185
 186 (defun inverse (a)
 187 "Args: (m)
 188 Returns the inverse of the the square matrix M; signals an error if M is ill
 189 conditioned or singular"
 190   (check-square-matrix a)
 191   (let ((n (num-rows a))
 192         (mode (max mode-re (la-data-mode a))))
 193     (declare (fixnum n))
 194     (let ((result (make-array (list n n) :initial-element 0)))
 195       (dotimes (i n)
 196                (declare (fixnum i))
 197                (setf (aref result i i) 1))
 198       (let ((mat (la-data-to-matrix a mode))
 199             (inv (la-data-to-matrix result mode))
 200             (iv (la-vector n mode-in))
 201             (v (la-vector n mode))
 202             (singular 0))
 203         (unwind-protect
 204             (progn
 205               (setf singular (lu-inverse-front mat n iv v mode inv))
 206               (la-matrix-to-data inv n n mode result))
 207           (la-free-matrix mat n)
 208           (la-free-matrix inv n)
 209           (la-free-vector iv)
 210           (la-free-vector v))
 211         (if (/= singular 0) (error "matrix is (numerically) singular"))
 212         result))))
 213
 214 ;;;;
 215 ;;;; SV Decomposition
 216 ;;;;
 217
 218 (defun sv-decomp (a)
 219 "Args: (a)
 220 A is a matrix of real numbers with at least as many rows as columns.
 221 Computes the singular value decomposition of A and returns a list of the form
 222 (U W V FLAG) where U and V are matrices whose columns are the left and right
 223 singular vectors of A and W is the sequence of singular values of A. FLAG is T
 224 if the algorithm converged, NIL otherwise."
 225   (check-matrix a)
 226   (let* ((m (num-rows a))
 227          (n (num-cols a))
 228          (mode (max mode-re (la-data-mode a)))
 229          (result (list (make-array (list m n))
 230                        (make-array n)
 231                        (make-array (list n n))
 232                        nil)))
 233     (if (< m n) (error "number of rows less than number of columns"))
 234     (if (= mode mode-cx) (error "complex SVD not available yet"))
 235     (let ((mat (la-data-to-matrix a mode))
 236           (w (la-vector n mode-re))
 237           (v (la-matrix n n mode-re))
 238           (converged 0))
 239       (unwind-protect
 240           (progn
 241             (setf converged (sv-decomp-front mat m n w v))
 242             (la-matrix-to-data mat m n mode (first result))
 243             (la-vector-to-data w n mode (second result))
 244             (la-matrix-to-data v n n mode (third result))
 245             (setf (fourth result) (if (/= 0.0 converged) t nil)))
 246         (la-free-matrix mat m)
 247         (la-free-vector w)
 248         (la-free-matrix v n))
 249       result)))
 250
 251
 252 ;;;;
 253 ;;;; QR Decomposition
 254 ;;;;
 255
 256 (defun qr-decomp (a &optional pivot)
 257 "Args: (a &optional pivot)
 258 A is a matrix of real numbers with at least as many rows as columns. Computes
 259 the QR factorization of A and returns the result in a list of the form (Q R).
 260 If PIVOT is true the columns of X are first permuted to place insure the
 261 absolute values of the diagonal elements of R are nonincreasing. In this case
 262 the result includes a third element, a list of the indices of the columns in
 263 the order in which they were used."
 264   (check-matrix a)
 265   (let* ((m (num-rows a))
 266          (n (num-cols a))
 267          (mode (max mode-re (la-data-mode a)))
 268          (p (if pivot 1 0))
 269          (result (if pivot
 270                      (list (make-array (list m n))
 271                            (make-array (list n n))
 272                            (make-array n))
 273                      (list (make-array (list m n)) (make-array (list n n))))))
 274     (if (< m n) (error "number of rows less than number of columns"))
 275     (if (= mode mode-cx) (error "complex QR decomposition not available yet"))
 276     (let ((mat (la-data-to-matrix a mode))
 277           (v (la-matrix n n mode-re))
 278           (jpvt (la-vector n mode-in)))
 279       (unwind-protect
 280           (progn
 281             (qr-decomp-front mat m n v jpvt p)
 282             (la-matrix-to-data mat m n mode (first result))
 283             (la-matrix-to-data v n n mode (second result))
 284             (if pivot (la-vector-to-data jpvt n mode-in (third result))))
 285         (la-free-matrix mat m)
 286         (la-free-matrix v n)
 287         (la-free-vector jpvt))
 288       result)))
 289
 290 ;;;;
 291 ;;;; Estimate of Condition Number for Lower Triangular Matrix
 292 ;;;;
 293
 294 (defun rcondest (a)
 295 "Args: (a)
 296 Returns an estimate of the reciprocal of the L1 condition number of an upper
 297 triangular matrix a."
 298   (check-square-matrix a)
 299   (let ((mode (max mode-re (la-data-mode a)))
 300         (n (num-rows a)))
 301     (if (= mode mode-cx)
 302         (error "complex condition estimate not available yet"))
 303     (let ((mat (la-data-to-matrix a mode))
 304           (est 0.0))
 305       (unwind-protect
 306           (setf est (rcondest-front mat n))
 307         (la-free-matrix mat n))
 308       est)))
 309
 310 ;;;;
 311 ;;;; Make Rotation Matrix
 312 ;;;;
 313
 314 (defun make-rotation (x y &optional alpha)
 315 "Args: (x y &optional alpha)
 316 Returns a rotation matrix for rotating from X to Y, or from X toward Y
 317 by angle ALPHA, in radians. X and Y are sequences of the same length."
 318   (check-sequence x)
 319   (check-sequence y)
 320   (if alpha (check-one-real alpha))
 321   (let* ((n (length x))
 322          (mode (max mode-re (la-data-mode x) (la-data-mode y)))
 323          (use-angle (if alpha 1 0))
 324          (angle (if alpha (float alpha 0.0) 0.0))
 325          (result (make-array (list n n))))
 326     (if (/= n (length y)) (error "sequences not the same length"))
 327     (if (= mode mode-cx) (error "complex data not supported yet"))
 328     (let ((px (la-data-to-vector x mode-re))
 329           (py (la-data-to-vector y mode-re))
 330           (rot (la-matrix n n mode-re)))
 331       (unwind-protect
 332           (progn
 333             (make-rotation-front n rot px py use-angle angle)
 334             (la-matrix-to-data rot n n mode-re result))
 335         (la-free-vector px)
 336         (la-free-vector py)
 337         (la-free-matrix rot n))
 338       result)))
 339
 340 ;;;;
 341 ;;;; Eigenvalues and Vectors
 342 ;;;;
 343
 344 (defun eigen (a)
 345 "Args: (a)
 346 Returns list of list of eigenvalues and list of eigenvectors of square,
 347 symmetric matrix A. Third element of result is NIL if algorithm converges.
 348 If the algorithm does not converge, the third element is an integer I.
 349 In this case the eigenvalues 0, ..., I are not reliable."
 350   (check-square-matrix a)
 351   (let ((mode (max mode-re (la-data-mode a)))
 352         (n (num-rows a)))
 353     (if (= mode mode-cx) (error "matrix must be real and symmetric"))
 354     (let ((evals (make-array n))
 355           (evecs (make-list (* n n)))
 356           (pa (la-data-to-vector (compound-data-seq a) mode-re))
 357           (w (la-vector n mode-re))
 358           (z (la-vector (* n n) mode-re))
 359           (fv1 (la-vector n mode-re))
 360           (ierr 0))
 361       (unwind-protect
 362           (progn
 363             (setf ierr (eigen-front pa n w z fv1))
 364             (la-vector-to-data w n mode-re evals)
 365             (la-vector-to-data z (* n n) mode-re evecs))
 366         (la-free-vector pa)
 367         (la-free-vector z)
 368         (la-free-vector w)
 369         (la-free-vector fv1))
 370       (list (nreverse evals)
 371             (nreverse (mapcar #'(lambda (x) (coerce x 'vector))
 372                               (split-list evecs n)))
 373             (if (/= 0 ierr) (- n ierr))))))
 374
 375 ;;;;
 376 ;;;; Spline Interpolation
 377 ;;;;
 378
 379 (defun make-smoother-args (x y xvals)
 380   (check-sequence x)
 381   (check-real x)
 382   (when y
 383         (check-sequence y)
 384         (check-real y))
 385   (unless (integerp xvals)
 386           (check-sequence xvals)
 387           (check-real xvals))
 388   (let* ((n (length x))
 389          (ns (if (integerp xvals) xvals (length xvals)))
 390          (result (list (make-list ns) (make-list ns))))
 391     (if (and y (/= n (length y))) (error "sequences not the same length"))
 392     (list x y n (if (integerp xvals) 0 1) ns xvals result)))
 393
 394 (defun get-smoother-result (args) (seventh args))
 395
 396 (defmacro with-smoother-data ((x y xvals is-reg) &rest body)
 397   `(progn
 398      (check-sequence ,x)
 399      (check-real ,x)
 400      (when ,is-reg
 401            (check-sequence ,y)
 402            (check-real ,y))
 403      (unless (integerp ,xvals)
 404              (check-sequence ,xvals)
 405              (check-real ,xvals))
 406      (let* ((supplied (not (integerp ,xvals)))
 407             (n (length ,x))
 408             (ns (if supplied (length ,xvals) ,xvals))
 409             (result (list (make-list ns) (make-list ns))))
 410        (if (and ,is-reg (/= n (length ,y)))
 411            (error "sequences not the same length"))
 412        (if (and (not supplied) (< ns 2))
 413            (error "too few points for interpolation"))
 414        (let* ((px (la-data-to-vector ,x mode-re))
 415               (py (if ,is-reg (la-data-to-vector ,y mode-re)))
 416               (pxs (if supplied
 417                        (la-data-to-vector ,xvals mode-re)
 418                        (la-vector ns mode-re)))
 419               (pys (la-vector ns mode-re)))
 420          (unless supplied (la-range-to-rseq n px ns pxs))
 421          (unwind-protect
 422              (progn ,@body
 423                     (la-vector-to-data pxs ns mode-re (first result))
 424                     (la-vector-to-data pys ns mode-re (second result)))
 425            (la-free-vector px)
 426            (if ,is-reg (la-free-vector py))
 427            (la-free-vector pxs)
 428            (la-free-vector pys))
 429          result))))
 430
 431 (defun spline (x y &key (xvals 30))
 432 "Args: (x y &key xvals)
 433 Returns list of x and y values of natural cubic spline interpolation of (X,Y).
 434 X must be strictly increasing. XVALS can be an integer, the number of equally
 435 spaced points to use in the range of X, or it can be a sequence of points at
 436 which to interpolate."
 437   (with-smoother-data (x y xvals t)
 438     (let ((work (la-vector (* 2 n) mode-re))
 439           (error 0))
 440       (unwind-protect
 441           (setf error (spline-front n px py ns pxs pys work))
 442         (la-free-vector work))
 443       (if (/= error 0) (error "bad data for splines")))))
 444
 445 ;;;;
 446 ;;;; Kernel Density Estimators and Smoothers
 447 ;;;;
 448
 449 (defun kernel-type-code (type)
 450   (cond ((eq type 'u) 0)
 451         ((eq type 't) 1)
 452         ((eq type 'g) 2)
 453         (t 3)))
 454
 455 (defun kernel-dens (x &key (type 'b) (width -1.0) (xvals 30))
 456 "Args: (x &key xvals width type)
 457 Returns list of x and y values of kernel density estimate of X. XVALS can be an
 458 integer, the number of equally spaced points to use in the range of X, or it
 459 can be a sequence of points at which to interpolate. WIDTH specifies the
 460 window width. TYPE specifies the lernel and should be one of the symbols G, T,
 461 U or B for gaussian, triangular, uniform or bisquare. The default is B."
 462   (check-one-real width)
 463   (with-smoother-data (x nil xvals nil) ;; warning about deleting unreachable code is TRUE -- 2nd arg=nil!
 464     (let ((code (kernel-type-code type))
 465           (error 0))
 466       (setf error (kernel-dens-front px n width pxs pys ns code))
 467       (if (/= 0 error) (error "bad kernel density data")))))
 468
 469 (defun kernel-smooth (x y &key (type 'b) (width -1.0) (xvals 30))
 470 "Args: (x y &key xvals width type)
 471 Returns list of x and y values of kernel smooth of (X,Y). XVALS can be an
 472 integer, the number of equally spaced points to use in the range of X, or it
 473 can be a sequence of points at which to interpolate. WIDTH specifies the
 474 window width. TYPE specifies the lernel and should be one of the symbols G, T,
 475 U or B for Gaussian, triangular, uniform or bisquare. The default is B."
 476   (check-one-real width)
 477   (with-smoother-data (x y xvals t)
 478     (let ((code (kernel-type-code type))
 479           (error 0))
 480       ;;(kernel-smooth-front px py n width pxs pys ns code)
 481       (kernel-smooth-Cport px py n width pxs pys ns code)
 482       (if (/= 0 error) (error "bad kernel density data")))))
 483
 484 (defun kernel-smooth-Cport (px py n width ;;wts wds ;; see above for mismatch?
 485                             xs ys ns ktype)
 486   "Port of kernel_smooth (Lib/kernel.c) to Lisp.
 487 FIXME:kernel-smooth-Cport"
 488   (cond ((< n 1) 1.0)
 489         ((and (< n 2) (<= width 0)) 1.0)
 490         (t (let* ((xmin (min px))
 491                   (xmax (max px))
 492                   (width (/ (- xmax xmin) (+ 1.0 (log n)))))
 493              (dotimes  (i (- ns 1))
 494                (setf (aref ys i)
 495                      (let ((wsum 0.0)
 496                            (ysum 0.0))
 497                        (dotimes (j (- n 1))
 498                          (let* ;; FIXME!?
 499                              ((lwidth (if wds (* width (aref wds j)) width))
 500                               (lwt (* (kernel-Cport (aref xs i) (aref px j) lwidth ktype) ;; px?
 501                                       (if wts (aref wts j) 1.0))))
 502                            (setf wsum (+ wsum lwt))
 503                            (setf ysum (if py (+ ysum (* lwt (aref py j))))))) ;; py? y?
 504                        (if py
 505                            (if (> wsum 0.0)
 506                                (/ ysum wsum)
 507                                0.0)
 508                            (/ wsum n)))))
 509              (values ys)))))
 510
 511 (defun kernel-Cport (x y w ktype)
 512   "Port of kernel() (Lib/kernel.c) to Lisp.
 513 x,y,w are doubles, type is an integer"
 514   (if (<= w 0.0)
 515       0.0
 516       (let ((z (- x y)))
 517         (cond ((eq ktype "B")
 518                (let* ((w (* w 2.0))
 519                       (z (* z 0.5)))
 520                  (if (and (> z -0.5)
 521                           (< z 0.5))
 522                      (/ (/ (* 15.0 (* (- 1.0 (* 4 z z))  ;; k/w
 523                                       (- 1.0 (* 4 z z)))) ;; k/w
 524                            8.0)
 525                         w)
 526                      0)))
 527               ((eq ktype "G")
 528                (let* ((w (* w 0.25))
 529                       (z (* z 4.0))
 530                       (k (/ (exp (* -0.5 z z))
 531                             (sqrt (* 2 PI)))))
 532                  (/ k w)))
 533               ((eq ktype "U")
 534                (let* ((w (* 1.5 w))
 535                       (z (* z 0.75))
 536                       (k (if (< (abs z) 0.5)
 537                              1.0
 538                              0.0)))
 539                  (/ k w)))
 540               ((eq ktype "T")
 541                (cond ((and (> z -1.0)
 542                            (< z 0.0))
 543                       (+ 1.0 z))  ;; k
 544                      ((and (> z 0.0)
 545                            (< z 1.0))
 546                       (- 1.0 z))  ;; k
 547                      (t 0.0)))
 548               (t (values 0.0))))))
 549
 550
 551 ;;;;
 552 ;;;; Lowess Smoother Interface
 553 ;;;;
 554
 555 (defun |base-lowess| (s1 s2 f nsteps delta)
 556   (check-sequence s1)
 557   (check-sequence s2)
 558   (check-real s1)
 559   (check-real s2)
 560   (check-one-real f)
 561   (check-one-fixnum nsteps)
 562   (check-one-real delta)
 563   (let* ((n (length s1))
 564          (result (make-list n)))
 565     (if (/= n (length s2)) (error "sequences not the same length"))
 566     (let ((x (la-data-to-vector s1 mode-re))
 567           (y (la-data-to-vector s2 mode-re))
 568           (ys (la-vector n mode-re))
 569           (rw (la-vector n mode-re))
 570           (res (la-vector n mode-re))
 571           (error 0))
 572       (unwind-protect
 573           (progn
 574             (setf error (base-lowess-front x y n f nsteps delta ys rw res))
 575             (la-vector-to-data ys n mode-re result))
 576         (la-free-vector x)
 577         (la-free-vector y)
 578         (la-free-vector ys)
 579         (la-free-vector rw)
 580         (la-free-vector res))
 581       (if (/= error 0) (error "bad data for lowess"))
 582       result)))
 583
 584 #|
 585 static LVAL add_contour_point(i, j, k, l, x, y, z, v, result)
 586         int i, j, k, l;
 587         RVector x, y;
 588         RMatrix z;
 589         double v;
 590         LVAL result;
 591 {
 592   LVAL pt;
 593   double p, q;
 594
 595   if ((z[i][j] <= v && v < z[k][l]) || (z[k][l] <= v && v < z[i][j])) {
 596     xlsave(pt);
 597     pt = mklist(2, NIL);
 598     p = (v - z[i][j]) / (z[k][l] - z[i][j]);
 599     q = 1.0 - p;
 600     rplaca(pt, cvflonum((FLOTYPE) (q * x[i] + p * x[k])));
 601     rplaca(cdr(pt), cvflonum((FLOTYPE) (q * y[j] + p * y[l])));
 602     result = cons(pt, result);
 603     xlpop();
 604   }
 605   return(result);
 606 }
 607
 608 LVAL xssurface_contour()
 609 {
 610   LVAL s1, s2, mat, result;
 611   RVector x, y;
 612   RMatrix z;
 613   double v;
 614   int i, j, n, m;
 615
 616   s1 = xsgetsequence();
 617   s2 = xsgetsequence();
 618   mat = xsgetmatrix();
 619   v = makedouble(xlgetarg());
 620   xllastarg();
 621
 622   n = seqlen(s1); m = seqlen(s2);
 623   if (n != numrows(mat) || m != numcols(mat)) xlfail("dimensions do not match");
 624   if (data_mode(s1) == CX || data_mode(s2) == CX || data_mode(mat) == CX)
 625     xlfail("data must be real");
 626
 627   x = (RVector) data_to_vector(s1, RE);
 628   y = (RVector) data_to_vector(s2, RE);
 629   z = (RMatrix) data_to_matrix(mat, RE);
 630
 631   xlsave1(result);
 632   result = NIL;
 633   for (i = 0; i < n - 1; i++) {
 634         for (j = 0; j < m - 1; j++) {
 635           result = add_contour_point(i, j, i, j+1, x, y, z, v, result);
 636           result = add_contour_point(i, j+1, i+1, j+1, x, y, z, v, result);
 637           result = add_contour_point(i+1, j+1, i+1, j, x, y, z, v, result);
 638           result = add_contour_point(i+1, j, i, j, x, y, z, v, result);
 639         }
 640   }
 641   xlpop();
 642
 643   free_vector(x);
 644   free_vector(y);
 645   free_matrix(z, n);
 646
 647   return(result);
 648 }
 649 |#
 650
 651 ;;;
 652 ;;; FFT
 653 ;;;
 654 ;;; FIXME:ajr
 655 ;;; replace with matlisp:fft  and matlisp:ifft (the latter for inverse mapping)
 656 ;;;
 657 (defun fft (x &optional inverse)
 658 "Args: (x &optional inverse)
 659 Returns unnormalized Fourier transform of X, or inverse transform if INVERSE
 660 is true."
 661   (check-sequence x)
 662   (let* ((n (length x))
 663          (mode (la-data-mode x))
 664          (isign (if inverse -1 1))
 665          (result (if (consp x) (make-list n) (make-array n))))
 666     (let ((px (la-data-to-vector x mode-cx))
 667           (work (la-vector (+ (* 4 n) 15) mode-re)))
 668       (unwind-protect
 669           (progn
 670             (fft-front n px work isign)
 671             (la-vector-to-data px n mode-cx result))
 672         (la-free-vector px)
 673         (la-free-vector work))
 674       result)))
 675
 676 ;;;
 677 ;;; SWEEP Operator: FIXME: use matlisp
 678 ;;;
 679
 680 (defun make-sweep-front (x y w n p mode has_w x_mean result)
 681   (declare (fixnum n p mode has_w))
 682   (let ((x_data nil)
 683         (result_data nil)
 684         (val 0.0)
 685         (dxi 0.0)
 686         (dyi 0.0)
 687         (dv 0.0)
 688         (dw 0.0)
 689         (sum_w 0.0)
 690         (dxik 0.0)
 691         (dxjk 0.0)
 692         (dyj 0.0)
 693         (dx_meani 0.0)
 694         (dx_meanj 0.0)
 695         (dy_mean 0.0)
 696         (has-w (if (/= 0 has_w) t nil))
 697         (RE 1))
 698     (declare (long-float val dxi dyi dv dw sum_w dxik dxjk dyj
 699                     dx_meani dx_meanj dy_mean))
 700 ;;    (declare-double val dxi dyi dv dw sum_w dxik dxjk dyj
 701 ;;                  dx_meani dx_meanj dy_mean)
 702
 703     (if (> mode RE) (error "not supported for complex data yet"))
 704
 705     (setf x_data (compound-data-seq x))
 706     (setf result_data (compound-data-seq result))
 707
 708     ;; find the mean of y
 709     (setf val 0.0)
 710     (setf sum_w 0.0)
 711     (dotimes (i n)
 712       (declare (fixnum i))
 713       (setf dyi (makedouble (aref y i)))
 714       (when has-w
 715         (setf dw (makedouble (aref w i)))
 716         (incf sum_w dw)
 717         (setf dyi (* dyi dw)))
 718       (incf val dyi))
 719     (if (not has-w) (setf sum_w (float n 0.0)))
 720     (if (<= sum_w 0.0) (error "non positive sum of weights"))
 721     (setf dy_mean (/ val sum_w))
 722
 723     ;; find the column means
 724     (dotimes (j p)
 725       (declare (fixnum j))
 726       (setf val 0.0)
 727       (dotimes (i n)
 728         (declare (fixnum i))
 729         (setf dxi (makedouble (aref x_data (+ (* p i) j))))
 730         (when has-w
 731           (setf dw (makedouble (aref w i)))
 732           (setf dxi (* dxi dw)))
 733         (incf val dxi))
 734       (setf (aref x_mean j) (/ val sum_w)))
 735
 736     ;; put 1/sum_w in topleft, means on left, minus means on top
 737     (setf (aref result_data 0) (/ 1.0 sum_w))
 738     (dotimes (i p)
 739       (declare (fixnum i))
 740       (setf dxi (makedouble (aref x_mean i)))
 741       (setf (aref result_data (+ i 1)) (- dxi))
 742       (setf (aref result_data (* (+ i 1) (+ p 2))) dxi))
 743     (setf (aref result_data (+ p 1)) (- dy_mean))
 744     (setf (aref result_data (* (+ p 1) (+ p 2))) dy_mean)
 745
 746     ;; put sums of adjusted cross products in body
 747     (dotimes (i p)
 748       (declare (fixnum i))
 749       (dotimes (j p)
 750         (declare (fixnum j))
 751         (setf val 0.0)
 752         (dotimes (k n)
 753           (declare (fixnum k))
 754           (setf dxik (makedouble (aref x_data (+ (* p k) i))))
 755           (setf dxjk (makedouble (aref x_data (+ (* p k) j))))
 756           (setf dx_meani (makedouble (aref x_mean i)))
 757           (setf dx_meanj (makedouble (aref x_mean j)))
 758           (setf dv (* (- dxik dx_meani) (- dxjk dx_meanj)))
 759           (when has-w
 760             (setf dw (makedouble (aref w k)))
 761             (setf dv (* dv dw)))
 762           (incf val dv))
 763         (setf (aref result_data (+ (* (+ i 1) (+ p 2)) (+ j 1))) val)
 764         (setf (aref result_data (+ (* (+ j 1) (+ p 2)) (+ i 1))) val))
 765       (setf val 0.0)
 766       (dotimes (j n)
 767         (declare (fixnum j))
 768         (setf dxik (makedouble (aref x_data (+ (* p j) i))))
 769         (setf dyj (makedouble (aref y j)))
 770         (setf dx_meani (makedouble (aref x_mean i)))
 771         (setf dv (* (- dxik dx_meani) (- dyj dy_mean)))
 772         (when has-w
 773           (setf dw (makedouble (aref w j)))
 774           (setf dv (* dv dw)))
 775         (incf val dv))
 776       (setf (aref result_data (+ (* (+ i 1) (+ p 2)) (+ p 1))) val)
 777       (setf (aref result_data (+ (* (+ p 1) (+ p 2)) (+ i 1))) val))
 778     (setf val 0.0)
 779     (dotimes (j n)
 780       (declare (fixnum j))
 781       (setf dyj (makedouble (aref y j)))
 782       (setf dv (* (- dyj dy_mean) (- dyj dy_mean)))
 783       (when has-w
 784         (setf dw (makedouble (aref w j)))
 785         (setf dv (* dv dw)))
 786       (incf val dv))
 787     (setf (aref result_data (+ (* (+ p 1) (+ p 2)) (+ p 1))) val)))
 788
 789 ;;; FIXME: use matlisp
 790 (defun sweep-in-place-front (a rows cols mode k tol)
 791   "Sweep algorithm for linear regression."
 792   (declare (long-float tol))
 793   (declare (fixnum rows cols mode k))
 794   (let ((data nil)
 795         (pivot 0.0)
 796         (aij 0.0)
 797         (aik 0.0)
 798         (akj 0.0)
 799         (akk 0.0)
 800         (RE 1))
 801     (declare (long-float pivot aij aik akj akk))
 802
 803     (if (> mode RE) (error "not supported for complex data yet"))
 804     (if (or (< k 0) (>= k rows) (>= k cols)) (error "index out of range"))
 805
 806     (setf tol (max tol machine-epsilon))
 807     (setf data (compound-data-seq a))
 808
 809     (setf pivot (makedouble (aref data (+ (* cols k) k))))
 810
 811     (cond
 812      ((or (> pivot tol) (< pivot (- tol)))
 813       (dotimes (i rows)
 814         (declare (fixnum i))
 815         (dotimes (j cols)
 816           (declare (fixnum j))
 817           (when (and (/= i k) (/= j k))
 818             (setf aij (makedouble (aref data (+ (* cols i) j))))
 819             (setf aik (makedouble (aref data (+ (* cols i) k))))
 820             (setf akj (makedouble (aref data (+ (* cols k) j))))
 821             (setf aij (- aij (/ (* aik akj) pivot)))
 822             (setf (aref data (+ (* cols i) j)) aij))))
 823
 824       (dotimes (i rows)
 825         (declare (fixnum i))
 826         (setf aik (makedouble (aref data (+ (* cols i) k))))
 827         (when (/= i k)
 828           (setf aik (/ aik pivot))
 829           (setf (aref data (+ (* cols i) k)) aik)))
 830
 831       (dotimes (j cols)
 832         (declare (fixnum j))
 833         (setf akj (makedouble (aref data (+ (* cols k) j))))
 834         (when (/= j k)
 835           (setf akj (- (/ akj pivot)))
 836           (setf (aref data (+ (* cols k) j)) akj)))
 837
 838       (setf akk (/ 1.0 pivot))
 839       (setf (aref data (+ (* cols k) k)) akk)
 840       1)
 841      (t 0))))
 842
 843 ;;; FIXME: use matlisp
 844 (defun make-sweep-matrix (x y &optional w)
 845 "Args: (x y &optional weights)
 846 X is a matrix, Y and WEIGHTS are sequences. Returns the sweep matrix for the
 847 (possibly weighted) regression of Y on X."
 848   (check-matrix x)
 849   (check-sequence y)
 850   (if w (check-sequence w))
 851   (let ((n (num-rows x))
 852         (p (num-cols x)))
 853     (if (/= n (length y)) (error "dimensions do not match"))
 854     (if (and w (/= n (length w))) (error "dimensions do not match"))
 855     (let ((mode (max (la-data-mode x)
 856                      (la-data-mode x)
 857                      (if w (la-data-mode w) 0)))
 858           (result (make-array (list (+ p 2) (+ p 2))))
 859           (x-mean (make-array p))
 860           (y (coerce y 'vector))
 861           (w (if w (coerce w 'vector)))
 862           (has-w (if w 1 0)))
 863       (make-sweep-front x y w n p mode has-w x-mean result)
 864       result)))
 865
 866 (defun sweep-in-place (a k tol)
 867   (check-matrix a)
 868   (check-one-fixnum k)
 869   (check-one-real tol)
 870   (let ((rows (num-rows a))
 871         (cols (num-cols a))
 872         (mode (la-data-mode a)))
 873     (let ((swept (sweep-in-place-front a rows cols mode k tol)))
 874       (if (/= 0 swept) t nil))))
 875
 876 (defun sweep-operator (a columns &optional tolerances)
 877 "Args: (a indices &optional tolerances)
 878 A is a matrix, INDICES a sequence of the column indices to be swept. Returns
 879 a list of the swept result and the list of the columns actually swept. (See
 880 MULTREG documentation.) If supplied, TOLERANCES should be a list of real
 881 numbers the same length as INDICES. An index will only be swept if its pivot
 882 element is larger than the corresponding element of TOLERANCES."
 883   (check-matrix a)
 884   (check-sequence columns)
 885   (if tolerances (check-sequence tolerances))
 886   (check-real a)
 887   (check-fixnum columns)
 888   (if tolerances (check-real tolerances))
 889   (do ((tol .0000001)
 890        (result (copy-array a))
 891        (swept-columns nil)
 892        (columns (coerce columns 'list) (cdr columns))
 893        (tolerances (if (consp tolerances) (coerce tolerances 'list))
 894                    (if (consp tolerances) (cdr tolerances))))
 895       ((null columns) (list result swept-columns))
 896       (let ((col (first columns))
 897             (tol (if (consp tolerances) (first tolerances) tol)))
 898         (if (sweep-in-place result col tol)
 899             (setf swept-columns (cons col swept-columns))))))
 900
 901
 902 ;;;
 903 ;;; AX+Y
 904 ;;;
 905
 906 ;;; matlisp:axpy
 907 ;;;
 908 (defun ax+y (a x y &optional lower)
 909 "Args (a x y &optional lower)
 910 Returns (+ (matmult A X) Y). If LOWER is not nil, A is taken to be lower
 911 triangular.
 912 This could probably be made more efficient."
 913   (check-square-matrix a)
 914   (check-sequence x)
 915   (check-sequence y)
 916   (check-real a)
 917   (check-real x)
 918   (check-real y)
 919   (let* ((n (num-rows a))
 920          (result (make-list n))
 921          (a (compound-data-seq a)))
 922     (declare (fixnum n))
 923     (if (or (/= n (length x)) (/= n (length y)))
 924         (error "dimensions do not match"))
 925     (do* ((tx (make-next-element x) (make-next-element x))
 926           (ty (make-next-element y))
 927           (tr (make-next-element result))
 928           (i 0 (+ i 1))
 929           (start 0 (+ start n))
 930           (end (if lower (+ i 1) n) (if lower (+ i 1) n)))
 931          ((<= n i) result)
 932        (declare (fixnum i start end))
 933        (let ((val (get-next-element ty i)))
 934          (dotimes (j end)
 935            (declare (fixnum j))
 936            (setf val (+ val (* (get-next-element tx j)
 937                                (aref a (+ start j))))))
 938          (set-next-element tr i val)))))
 939
 940 ;;;;
 941 ;;;; Maximization and Numerical Derivatives
 942 ;;;;
 943
 944 (defvar *maximize-callback-function* nil)
 945 (defvar *maximize-callback-arg* nil)
 946
 947 (defun data2double (n data ptr)
 948   (declare (fixnum n))
 949   (let* ((seq (compound-data-seq data))
 950          (elem (make-next-element seq)))
 951     (if (/= (length seq) n) (error "bad data size"))
 952     (dotimes (i n)
 953       (declare (fixnum i))
 954       (la-put-double ptr i (get-next-element elem i)))))
 955
 956 (defun maximize-callback (n px pfval pgrad phess pderivs)
 957   (la-vector-to-data px n mode-re *maximize-callback-arg*)
 958   (let* ((val (funcall *maximize-callback-function* *maximize-callback-arg*))
 959          (derivs (if (consp val) (- (length val) 1) 0)))
 960     (la-put-integer pderivs 0 derivs)
 961     (la-put-double pfval 0 (if (consp val) (first val) val))
 962     (if (<= 1 derivs) (data2double n (second val) pgrad))
 963     (if (<= 2 derivs) (data2double (* n n) (third val) phess))))
 964
 965 (defun numgrad (f x &optional scale (h -1.0))
 966 "Args: (f x &optional scale derivstep)
 967 Computes the numerical gradient of F at X."
 968   (check-sequence x)
 969   (check-real x)
 970   (when scale
 971         (check-sequence scale)
 972         (check-real scale))
 973   (check-one-real h)
 974   (let* ((n (length x))
 975          (result (make-list n)))
 976     (if (and scale (/= n (length scale)))
 977         (error "scale not the same length as x"))
 978     (let ((*maximize-callback-function* f)
 979           (*maximize-callback-arg* (make-list n)))
 980       (let ((px (la-data-to-vector x mode-re))
 981             (pgrad (la-vector n mode-re))
 982             (pscale (la-data-to-vector
 983                      (if scale scale (make-list n :initial-element 1.0))
 984                      mode-re)))
 985         (unwind-protect
 986             (progn
 987               (numgrad-front n px pgrad h pscale)
 988               (la-vector-to-data pgrad n mode-re result))
 989           (la-free-vector px)
 990           (la-free-vector pgrad)
 991           (la-free-vector pscale))))
 992       result))
 993
 994 (defun numhess (f x &optional scale (h -1.0) all)
 995 "Args: (f x &optional scale derivstep)
 996 Computes the numerical Hessian matrix of F at X."
 997   (check-sequence x)
 998   (check-real x)
 999   (when scale
1000         (check-sequence scale)
1001         (check-real scale))
1002   (check-one-real h)
1003   (let* ((n (length x))
1004          (result (if all
1005                      (list nil (make-list n) (make-array (list n n)))
1006                      (make-array (list n n)))))
1007     (if (and scale (/= n (length scale)))
1008         (error "scale not the same length as x"))
1009     (let ((*maximize-callback-function* f)
1010           (*maximize-callback-arg* (make-list n)))
1011       (let ((hess-data (compound-data-seq (if all (third result) result)))
1012             (px (la-data-to-vector x mode-re))
1013             (pf (la-vector 1 mode-re))
1014             (pgrad (la-vector n mode-re))
1015             (phess (la-vector (* n n) mode-re))
1016             (pscale (la-data-to-vector
1017                      (if scale scale (make-list n :initial-element 1.0))
1018                      mode-re)))
1019         (unwind-protect
1020             (progn
1021               (numhess-front n px pf pgrad phess h pscale)
1022               (when all
1023                     (setf (first result) (la-get-double pf 0))
1024                     (la-vector-to-data pgrad n mode-re (second result)))
1025               (la-vector-to-data phess (* n n) mode-re hess-data))
1026           (la-free-vector pf)
1027           (la-free-vector px)
1028           (la-free-vector pgrad)
1029           (la-free-vector phess)
1030           (la-free-vector pscale))))
1031     result))
1032
1033 (defun init-minfo-ipar-values (n ipars)
1034   (let* ((TRUE 1)
1035          (FALSE 0)
1036          (k 0)
1037          (m 0)
1038          (itnlimit -1)
1039          (backtrack TRUE)
1040          (verbose 0)
1041          (vals_suppl FALSE)
1042          (exptilt TRUE)
1043          (count 0)
1044          (termcode 0))
1045     (setf (aref ipars 0) n)
1046     (setf (aref ipars 1) m)
1047     (setf (aref ipars 2) k)
1048     (setf (aref ipars 3) itnlimit)
1049     (setf (aref ipars 4) backtrack)
1050     (setf (aref ipars 5) verbose)
1051     (setf (aref ipars 6) vals_suppl)
1052     (setf (aref ipars 7) exptilt)
1053     (setf (aref ipars 8) count)
1054     (setf (aref ipars 9) termcode)))
1055
1056 (defun init-minfo-dpar-values (h dpars)
1057   (let ((typf 1.0)
1058         (gradtol -1.0)
1059         (steptol -1.0)
1060         (maxstep -1.0)
1061         (dflt 0.0)
1062         (tilt 0.0)
1063         (newtilt 0.0)
1064         (hessadd 0.0))
1065     (setf (aref dpars 0) typf)
1066     (setf (aref dpars 1) h)
1067     (setf (aref dpars 2) gradtol)
1068     (setf (aref dpars 3) steptol)
1069     (setf (aref dpars 4) maxstep)
1070     (setf (aref dpars 5) dflt)
1071     (setf (aref dpars 6) tilt)
1072     (setf (aref dpars 7) newtilt)
1073     (setf (aref dpars 8) hessadd)))
1074
1075 (defun init-minfo-internals (n h internals)
1076   (let ((ipars (aref internals 8))
1077         (dpars (aref internals 9)))
1078     (init-minfo-ipar-values n ipars)
1079     (init-minfo-dpar-values h dpars)))
1080
1081 (defun new-minfo-internals (f x &key scale ((:derivstep h) -1.0))
1082   (check-sequence x)
1083   (check-real x)
1084   (check-one-real h)
1085   (let ((n (length x)))
1086     (when scale
1087           (check-sequence scale)
1088           (check-real scale)
1089           (if (/= n (length scale)) (error "scale and x not the same length")))
1090     (let ((internals (make-array 12)))
1091       (setf (aref internals 0) f)
1092       (setf (aref internals 3) (if (consp x) (copy-list x) (coerce x 'list)))
1093       (setf (aref internals 4)
1094             (if scale (copy-seq scale) (make-array n :initial-element 1.0)))
1095       (setf (aref internals 5) (make-list (+ 1 n (* n n))))
1096       (setf (aref internals 8) (make-array 10))
1097       (setf (aref internals 9) (make-array 9))
1098       (init-minfo-internals n h internals)
1099       internals)))
1100
1101 (defun minfo-maximize (internals &optional verbose)
1102   "This function does what?"
1103   (let* ((f (aref internals 0))
1104          (x (aref internals 3))
1105          (fvals (aref internals 5))
1106          (n (length x))
1107          (v (if verbose (if (integerp verbose) verbose 1) -1)))
1108     (setf (aref internals 3) (copy-list x))
1109     (setf (aref internals 5) (copy-list fvals))
1110     (let ((*maximize-callback-function* f)
1111           (*maximize-callback-arg* (make-list n)))
1112       (let* ((x (aref internals 3))
1113              (scale (aref internals 4))
1114              (fvals (aref internals 5))
1115              (ip (aref internals 8))
1116              (dp (aref internals 9))
1117              (px (la-data-to-vector x mode-re))
1118              (pscale (la-data-to-vector scale mode-re))
1119              (pfvals (la-vector (length fvals) mode-re))
1120              (pip (la-data-to-vector ip mode-in))
1121              (pdp (la-data-to-vector dp mode-re)))
1122         (unwind-protect
1123             (progn
1124               (base-minfo-maximize px pfvals pscale pip pdp v)) ;; access to C
1125           (la-vector-to-data px n mode-re x)
1126           (la-vector-to-data pfvals (+ 1 n (* n n)) mode-re fvals)
1127           (la-vector-to-data pip (length ip) mode-in ip)
1128           (la-vector-to-data pdp (length dp) mode-re dp))
1129         (get-buf)))))
1130
1131 ;;;;
1132 ;;;; Miscellaneous Routines
1133 ;;;;
1134
1135
1136 (defun split-list (x n)
1137 "Args: (list cols)
1138 Returns a list of COLS lists of equal length of the elements of LIST.
1139 Example: (split-list '(1 2 3 4 5 6) 2) returns ((1 2 3) (4 5 6))"
1140   (check-one-fixnum n)
1141   (if (/= (rem (length x) n) 0) (error "length not divisible by ~a" n))
1142   (flet ((next-split ()
1143            (let ((result nil)
1144                  (end nil))
1145              (dotimes (i n result)
1146                (declare (fixnum i))
1147                (let ((c-elem (list (first x))))
1148                  (cond ((null result)
1149                         (setf result c-elem)
1150                         (setf end result))
1151                        (t
1152                         (setf (rest end) c-elem)
1153                         (setf end (rest end)))))
1154                (setf x (rest x))))))
1155     (let ((result nil)
1156           (end nil)
1157           (k (/ (length x) n)))
1158       (declare (fixnum k))
1159       (dotimes (i k result)
1160         (declare (fixnum i))
1161         (let ((c-sub (list (next-split))))
1162           (cond ((null result)
1163                  (setf result c-sub)
1164                  (setf end result))
1165                 (t
1166                  (setf (rest end) c-sub)
1167                  (setf end (rest end)))))))))