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