qr.lisp 11.3 KB
 liam committed Feb 18, 2008 1 2 ``````;; QR decomposition ;; Liam Healy 2008-02-17 11:05:20EST qr.lisp `````` Liam Healy committed Dec 22, 2009 3 ``````;; Time-stamp: <2009-12-22 22:34:28EST qr.lisp> `````` lhealy committed Jul 24, 2008 4 ``````;; \$Id\$ `````` liam committed Apr 28, 2006 5 6 7 `````` (in-package :gsl) `````` Liam Healy committed Sep 20, 2009 8 9 ``````;;; /usr/include/gsl/gsl_linalg.h `````` liam committed Feb 18, 2008 10 11 12 13 14 ``````;;; FDL ;;; A general rectangular M-by-N matrix A has a ;;; QR decomposition into the product of an orthogonal ;;; M-by-M square matrix Q (where Q^T Q = I) and ;;; an M-by-N right-triangular matrix R, A = Q R. `````` liam committed Apr 28, 2006 15 `````` `````` liam committed Feb 18, 2008 16 17 18 19 20 21 ``````;;; This decomposition can be used to convert the linear system A x = b ;;; into the triangular system R x = Q^T b, which can be solved by ;;; back-substitution. Another use of the QR decomposition is to ;;; compute an orthonormal basis for a set of vectors. The first N ;;; columns of Q form an orthonormal basis for the range of A, ;;; ran(A), when A has full column rank. `````` liam committed Apr 28, 2006 22 `````` `````` Liam Healy committed Sep 20, 2009 23 24 25 26 ``````(defmfun QR-decomposition (A &optional (tau (make-marray 'double-float :dimensions (min (dim0 A) (dim1 A))))) `````` liam committed Apr 28, 2006 27 `````` "gsl_linalg_QR_decomp" `````` Liam Healy committed Aug 10, 2008 28 29 30 `````` (((mpointer A) :pointer) ((mpointer tau) :pointer)) :inputs (A) :outputs (A tau) `````` liam committed Feb 18, 2008 31 32 33 `````` :documentation ; FDL "Factorize the M-by-N matrix A into the QR decomposition A = Q R. On output the diagonal and `````` liam committed Apr 28, 2006 34 `````` upper triangular part of the input matrix contain the matrix `````` liam committed Feb 18, 2008 35 36 37 38 39 `````` R. The vector tau and the columns of the lower triangular part of the matrix A contain the Householder coefficients and Householder vectors which encode the orthogonal matrix Q. The vector tau must be of length k=min(M,N). The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1 `````` Liam Healy committed Aug 10, 2008 40 `````` where Q_i = I - tau_i v_i v_i^T and v_i is the `````` liam committed Feb 18, 2008 41 42 `````` Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same storage scheme as used by lapack. `````` liam committed Apr 28, 2006 43 44 `````` The algorithm used to perform the decomposition is Householder QR (Golub `````` liam committed Feb 18, 2008 45 `````` & Van Loan, Matrix Computations, Algorithm 5.2.1).") `````` liam committed Apr 28, 2006 46 `````` `````` Liam Healy committed Feb 23, 2009 47 48 49 50 51 52 53 54 55 56 57 58 ``````(defmfun QR-solve (QR tau b &optional x-spec &aux (x (if (eq x-spec t) (make-marray 'double-float :dimensions (dimensions b)) x-spec))) ("gsl_linalg_QR_svx" "gsl_linalg_QR_solve") ((((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer b) :pointer)) (((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer b) :pointer) ((mpointer x) :pointer))) :inputs (QR tau b x) `````` Liam Healy committed Aug 10, 2008 59 `````` :outputs (x) `````` Liam Healy committed Feb 23, 2009 60 `````` :return ((or x b)) `````` liam committed Feb 18, 2008 61 `````` :documentation ; FDL `````` Liam Healy committed Feb 23, 2009 62 63 64 65 66 67 68 69 70 71 `````` "Solve the square system A x = b using the QR decomposition of A into (QR, tau) given by QR-decomp. The least-squares solution for rectangular systems can be found using QR-lssolve. If x-spec is NIL (default), the solution will replace b. If x-spec is T, then an array will be created and the solution returned in it. If x-spec is a marray, the solution will be returned in it. If x-spec is non-NIL, on output the solution is stored in x and b is not modified. The solution is returned from the function call.") (defmfun QR-solve-least-squares `````` Liam Healy committed Sep 20, 2009 72 73 74 75 `````` (QR tau b &optional (x (make-marray 'double-float :dimensions (dim1 QR))) (residual (make-marray 'double-float :dimensions (dim0 QR)))) `````` liam committed Apr 28, 2006 76 `````` "gsl_linalg_QR_lssolve" `````` Liam Healy committed Aug 10, 2008 77 78 79 80 81 `````` (((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer b) :pointer) ((mpointer x) :pointer) ((mpointer residual) :pointer)) :inputs (QR tau b) :outputs (x residual) `````` liam committed Feb 18, 2008 82 `````` :documentation ; FDL `````` Liam Healy committed Aug 10, 2008 83 84 85 86 87 88 89 90 `````` "The least squares solution to the overdetermined system A x = b where the matrix A has more rows than columns. The least squares solution minimizes the Euclidean norm of the residual, ||Ax - b||.The routine uses the QR decomposition of A into (QR, tau) given by #'QR-decomposition. The solution is returned in x. The residual is computed as a by-product and stored in residual.") (defmfun QR-QTvector (QR tau v) `````` liam committed Jul 04, 2006 91 `````` "gsl_linalg_QR_QTvec" `````` Liam Healy committed Aug 10, 2008 92 93 94 95 `````` (((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer v) :pointer)) :inputs (QR tau) :outputs (v) `````` liam committed Feb 18, 2008 96 97 98 99 100 101 102 `````` :documentation ; FDL "Apply the matrix Q^T encoded in the decomposition (QR, tau) to the vector v, storing the result Q^T v in v. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.") `````` Liam Healy committed Aug 10, 2008 103 ``````(defmfun QR-Qvector (QR tau v) `````` liam committed Jul 04, 2006 104 `````` "gsl_linalg_QR_Qvec" `````` Liam Healy committed Aug 10, 2008 105 106 107 108 `````` (((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer v) :pointer)) :inputs (QR tau) :outputs (v) `````` liam committed Feb 18, 2008 109 110 111 112 `````` :documentation ; FDL "Apply the matrix Q encoded in the decomposition (QR, tau) to the vector v, storing the result Q v in v. The matrix multiplication is carried out directly using `````` liam committed Apr 28, 2006 113 `````` the encoding of the Householder vectors without needing to form the full `````` liam committed Feb 18, 2008 114 `````` matrix Q.") `````` liam committed Apr 28, 2006 115 `````` `````` Liam Healy committed Feb 23, 2009 116 117 118 119 120 121 122 123 124 125 126 127 ``````(defmfun QR-Rsolve (QR b &optional x-spec &aux (x (if (eq x-spec t) (make-marray 'double-float :dimensions (dimensions b)) x-spec))) ("gsl_linalg_QR_Rsvx" "gsl_linalg_QR_Rsolve") ((((mpointer QR) :pointer) ((mpointer b) :pointer)) (((mpointer QR) :pointer) ((mpointer b) :pointer) ((mpointer x) :pointer))) :inputs (QR b x) :outputs (b x) :return ((or x b)) `````` liam committed Feb 18, 2008 128 `````` :documentation ; FDL `````` Liam Healy committed Aug 10, 2008 129 `````` "Solve the triangular system R x = b for x. It may be useful if the `````` Liam Healy committed Feb 23, 2009 130 131 132 133 134 135 `````` product b' = Q^T b has already been computed using QR-QTvec. If x-spec is NIL (default), the solution will replace b. If x-spec is T, then an array will be created and the solution returned in it. If x-spec is a marray, the solution will be returned in it. If x-spec is non-NIL, on output the solution is stored in x and b is not modified. The solution is returned from the function call.") `````` liam committed Apr 28, 2006 136 `````` `````` Liam Healy committed Sep 20, 2009 137 138 139 140 141 ``````(defmfun QR-unpack (QR tau &optional (Q (make-marray 'double-float :dimensions (list (dim0 QR) (dim0 QR)))) (R (make-marray 'double-float :dimensions (dimensions QR)))) `````` liam committed Apr 28, 2006 142 `````` "gsl_linalg_QR_unpack" `````` Liam Healy committed Aug 10, 2008 143 144 145 146 `````` (((mpointer QR) :pointer) ((mpointer tau) :pointer) ((mpointer Q) :pointer) ((mpointer R) :pointer)) :inputs (QR tau) :outputs (Q R) `````` liam committed Feb 18, 2008 147 148 149 150 `````` :documentation ; FDL "Unpack the encoded QR decomposition (QR, tau) into the matrices Q and R where Q is M-by-M and R is M-by-N.") `````` liam committed Apr 28, 2006 151 `````` `````` Liam Healy committed Sep 20, 2009 152 153 ``````(defmfun QR-QRsolve (Q R b &optional (x (make-marray 'double-float :dimensions (dim0 b)))) `````` liam committed Apr 28, 2006 154 `````` "gsl_linalg_QR_QRsolve" `````` Liam Healy committed Aug 10, 2008 155 156 157 158 `````` (((mpointer Q) :pointer) ((mpointer R) :pointer) ((mpointer b) :pointer) ((mpointer x) :pointer)) :inputs (Q R b) :outputs (x) `````` liam committed Feb 18, 2008 159 160 161 162 `````` :documentation ; FDL "Solves the system R x = Q^T b for x. It can be used when the QR decomposition of a matrix is available in unpacked form as Q, R).") `````` liam committed Apr 28, 2006 163 `````` `````` liam committed Feb 18, 2008 164 ``````(defmfun QR-update (Q R w v) `````` liam committed Apr 28, 2006 165 `````` "gsl_linalg_QR_update" `````` Liam Healy committed Aug 10, 2008 166 167 168 169 `````` (((mpointer Q) :pointer) ((mpointer R) :pointer) ((mpointer w) :pointer) ((mpointer v) :pointer)) :inputs (Q R w v) :outputs (w Q R) `````` liam committed Feb 18, 2008 170 171 172 173 174 175 176 177 `````` :return (Q R) :documentation ; FDL "Perform a rank-1 update w v^T of the QR decomposition (Q, R). The update is given by Q'R' = Q R + w v^T where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update.") `````` Liam Healy committed Feb 23, 2009 178 179 180 181 182 183 184 185 186 187 ``````(defmfun R-solve (R b &optional x-spec &aux (x (if (eq x-spec t) (make-marray 'double-float :dimensions (dimensions b)) x-spec))) ("gsl_linalg_R_svx" "gsl_linalg_R_solve") ((((mpointer R) :pointer) ((mpointer b) :pointer)) (((mpointer R) :pointer) ((mpointer b) :pointer) ((mpointer x) :pointer))) :inputs (R b x) `````` Liam Healy committed Aug 10, 2008 188 `````` :outputs (x) `````` Liam Healy committed Feb 23, 2009 189 `````` :return ((or x b)) `````` liam committed Feb 18, 2008 190 `````` :documentation ; FDL `````` Liam Healy committed Feb 23, 2009 191 192 193 194 195 196 197 198 `````` "Solve the triangular system R x = b in-place. On input x should contain the right-hand side b, which is replaced by the solution on output. If x-spec is NIL (default), the solution will replace b. If x-spec is T, then an array will be created and the solution returned in it. If x-spec is a marray, the solution will be returned in it. If x-spec is non-NIL, on output the solution is stored in x and b is not modified. The solution is returned from the function call.") `````` Liam Healy committed Sep 20, 2009 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 `````` ;;; Examples and unit test, from linalg/test.c (defun test-qr-solve-dim (matrix) "Solve the linear equation using QR with the supplied matrix and a right-hand side vector which is the reciprocal of one more than the index." (let ((dim (dim0 matrix))) (multiple-value-bind (QR tau) (QR-decomposition (copy matrix)) (QR-solve QR tau (create-rhs-vector dim) T)))) (defun test-qr-qrsolve-dim (matrix) "Solve the linear equation using QR with the supplied matrix and a right-hand side vector which is the reciprocal of one more than the index." (let ((dim (dim0 matrix))) (multiple-value-bind (QR tau) (QR-decomposition (copy matrix)) (multiple-value-bind (Q R) (QR-unpack QR tau) (QR-QRsolve Q R (create-rhs-vector dim)))))) (defun test-qr-lssolve-dim (matrix) "Solve the linear equation using QR least squares with the supplied matrix and a right-hand side vector which is the reciprocal of one more than the index. Returns the solution and the residual." (let ((dim (dim0 matrix))) (multiple-value-bind (QR tau) (QR-decomposition (copy matrix)) ;; Residual not checked. (QR-solve-least-squares QR tau (create-rhs-vector dim))))) (defun test-qr-decomp-dim (matrix) "Solve the QR decomposition with the supplied matrix and a right-hand side vector which is the reciprocal of one more than the index." (multiple-value-bind (QR tau) (QR-decomposition (copy matrix)) (multiple-value-bind (Q R) (QR-unpack QR tau) (matrix-product Q R)))) (defun test-qr-update-dim (matrix) "Test QR rank-1 update; this should return a matrix with all elements near zero." (let* ((dim0 (dim0 matrix)) (dim1 (dim1 matrix)) `````` Liam Healy committed Dec 22, 2009 246 247 `````` (u (create-matrix (lambda (i) (sin (1+ i))) dim0 nil)) (v (create-matrix `````` Liam Healy committed Sep 20, 2009 248 `````` (lambda (i) (+ (cos (+ 2 i)) (sin (+ 3 (expt i 2))))) `````` Liam Healy committed Dec 22, 2009 249 `````` dim1 nil)) `````` Liam Healy committed Sep 20, 2009 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 `````` (qr1 (create-matrix (lambda (i j) (+ (maref matrix i j) (* (maref u i) (maref v j)))) dim0 dim1)) (qr2 (copy matrix)) (w (make-marray 'double-float :dimensions dim0))) (multiple-value-bind (QR2 tau) (QR-decomposition qr2) (multiple-value-bind (Q2 R2) (QR-unpack QR2 tau) ;; compute w = Q^T u (matrix-product Q2 u w 1.0d0 0.0d0 :trans) (QR-update Q2 R2 w v) (matrix-product Q2 R2 qr2 1.0d0 0.0d0) (elt- qr1 qr2))))) (save-test qr ;; test_QR_solve (test-qr-solve-dim *hilb2*) (test-qr-solve-dim *hilb3*) (test-qr-solve-dim *hilb4*) (test-qr-solve-dim *hilb12*) (test-qr-solve-dim *vander2*) (test-qr-solve-dim *vander3*) (test-qr-solve-dim *vander4*) (test-qr-solve-dim *vander12*) ;; test_QR_QRsolve (test-qr-qrsolve-dim *hilb2*) (test-qr-qrsolve-dim *hilb3*) (test-qr-qrsolve-dim *hilb4*) (test-qr-qrsolve-dim *hilb12*) (test-qr-qrsolve-dim *vander2*) (test-qr-qrsolve-dim *vander3*) (test-qr-qrsolve-dim *vander4*) (test-qr-qrsolve-dim *vander12*) ;; test_QR_lssolve (test-qr-lssolve-dim *m53*) (test-qr-lssolve-dim *hilb2*) (test-qr-lssolve-dim *hilb3*) (test-qr-lssolve-dim *hilb4*) (test-qr-lssolve-dim *hilb12*) (test-qr-lssolve-dim *vander2*) (test-qr-lssolve-dim *vander3*) (test-qr-lssolve-dim *vander4*) (test-qr-lssolve-dim *vander12*) ;; test_QR_decomp (test-qr-decomp-dim *m35*) (test-qr-decomp-dim *m53*) (test-qr-decomp-dim *hilb2*) (test-qr-decomp-dim *hilb3*) (test-qr-decomp-dim *hilb4*) (test-qr-decomp-dim *hilb12*) (test-qr-decomp-dim *vander2*) (test-qr-decomp-dim *vander3*) (test-qr-decomp-dim *vander4*) (test-qr-decomp-dim *vander12*) ;; test_QR_update (test-qr-update-dim *m35*) (test-qr-update-dim *m53*) (test-qr-update-dim *hilb2*) (test-qr-update-dim *hilb3*) (test-qr-update-dim *hilb4*) (test-qr-update-dim *hilb12*) (test-qr-update-dim *vander2*) (test-qr-update-dim *vander3*) (test-qr-update-dim *vander4*) (test-qr-update-dim *vander12*))``````