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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE DGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, 00002 + LDV, WORK, LWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.3.0) -- 00005 * 00006 * -- Contributed by Zlatko Drmac of the University of Zagreb and -- 00007 * -- Kresimir Veselic of the Fernuniversitaet Hagen -- 00008 * November 2010 00009 * 00010 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00011 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00012 * 00013 * This routine is also part of SIGMA (version 1.23, October 23. 2008.) 00014 * SIGMA is a library of algorithms for highly accurate algorithms for 00015 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the 00016 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. 00017 * 00018 IMPLICIT NONE 00019 * .. 00020 * .. Scalar Arguments .. 00021 INTEGER INFO, LDA, LDV, LWORK, M, MV, N 00022 CHARACTER*1 JOBA, JOBU, JOBV 00023 * .. 00024 * .. Array Arguments .. 00025 DOUBLE PRECISION A( LDA, * ), SVA( N ), V( LDV, * ), 00026 + WORK( LWORK ) 00027 * .. 00028 * 00029 * Purpose 00030 * ======= 00031 * 00032 * DGESVJ computes the singular value decomposition (SVD) of a real 00033 * M-by-N matrix A, where M >= N. The SVD of A is written as 00034 * [++] [xx] [x0] [xx] 00035 * A = U * SIGMA * V^t, [++] = [xx] * [ox] * [xx] 00036 * [++] [xx] 00037 * where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal 00038 * matrix, and V is an N-by-N orthogonal matrix. The diagonal elements 00039 * of SIGMA are the singular values of A. The columns of U and V are the 00040 * left and the right singular vectors of A, respectively. 00041 * 00042 * Further Details 00043 * ~~~~~~~~~~~~~~~ 00044 * The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane 00045 * rotations. The rotations are implemented as fast scaled rotations of 00046 * Anda and Park [1]. In the case of underflow of the Jacobi angle, a 00047 * modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses 00048 * column interchanges of de Rijk [2]. The relative accuracy of the computed 00049 * singular values and the accuracy of the computed singular vectors (in 00050 * angle metric) is as guaranteed by the theory of Demmel and Veselic [3]. 00051 * The condition number that determines the accuracy in the full rank case 00052 * is essentially min_{D=diag} kappa(A*D), where kappa(.) is the 00053 * spectral condition number. The best performance of this Jacobi SVD 00054 * procedure is achieved if used in an accelerated version of Drmac and 00055 * Veselic [5,6], and it is the kernel routine in the SIGMA library [7]. 00056 * Some tunning parameters (marked with [TP]) are available for the 00057 * implementer. 00058 * The computational range for the nonzero singular values is the machine 00059 * number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even 00060 * denormalized singular values can be computed with the corresponding 00061 * gradual loss of accurate digits. 00062 * 00063 * Contributors 00064 * ~~~~~~~~~~~~ 00065 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) 00066 * 00067 * References 00068 * ~~~~~~~~~~ 00069 * [1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling. 00070 * SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174. 00071 * [2] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the 00072 * singular value decomposition on a vector computer. 00073 * SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371. 00074 * [3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR. 00075 * [4] Z. Drmac: Implementation of Jacobi rotations for accurate singular 00076 * value computation in floating point arithmetic. 00077 * SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222. 00078 * [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. 00079 * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. 00080 * LAPACK Working note 169. 00081 * [6] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. 00082 * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. 00083 * LAPACK Working note 170. 00084 * [7] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, 00085 * QSVD, (H,K)-SVD computations. 00086 * Department of Mathematics, University of Zagreb, 2008. 00087 * 00088 * Bugs, Examples and Comments 00089 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~ 00090 * Please report all bugs and send interesting test examples and comments to 00091 * drmac@math.hr. Thank you. 00092 * 00093 * Arguments 00094 * ========= 00095 * 00096 * JOBA (input) CHARACTER* 1 00097 * Specifies the structure of A. 00098 * = 'L': The input matrix A is lower triangular; 00099 * = 'U': The input matrix A is upper triangular; 00100 * = 'G': The input matrix A is general M-by-N matrix, M >= N. 00101 * 00102 * JOBU (input) CHARACTER*1 00103 * Specifies whether to compute the left singular vectors 00104 * (columns of U): 00105 * = 'U': The left singular vectors corresponding to the nonzero 00106 * singular values are computed and returned in the leading 00107 * columns of A. See more details in the description of A. 00108 * The default numerical orthogonality threshold is set to 00109 * approximately TOL=CTOL*EPS, CTOL=DSQRT(M), EPS=DLAMCH('E'). 00110 * = 'C': Analogous to JOBU='U', except that user can control the 00111 * level of numerical orthogonality of the computed left 00112 * singular vectors. TOL can be set to TOL = CTOL*EPS, where 00113 * CTOL is given on input in the array WORK. 00114 * No CTOL smaller than ONE is allowed. CTOL greater 00115 * than 1 / EPS is meaningless. The option 'C' 00116 * can be used if M*EPS is satisfactory orthogonality 00117 * of the computed left singular vectors, so CTOL=M could 00118 * save few sweeps of Jacobi rotations. 00119 * See the descriptions of A and WORK(1). 00120 * = 'N': The matrix U is not computed. However, see the 00121 * description of A. 00122 * 00123 * JOBV (input) CHARACTER*1 00124 * Specifies whether to compute the right singular vectors, that 00125 * is, the matrix V: 00126 * = 'V' : the matrix V is computed and returned in the array V 00127 * = 'A' : the Jacobi rotations are applied to the MV-by-N 00128 * array V. In other words, the right singular vector 00129 * matrix V is not computed explicitly, instead it is 00130 * applied to an MV-by-N matrix initially stored in the 00131 * first MV rows of V. 00132 * = 'N' : the matrix V is not computed and the array V is not 00133 * referenced 00134 * 00135 * M (input) INTEGER 00136 * The number of rows of the input matrix A. M >= 0. 00137 * 00138 * N (input) INTEGER 00139 * The number of columns of the input matrix A. 00140 * M >= N >= 0. 00141 * 00142 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00143 * On entry, the M-by-N matrix A. 00144 * On exit : 00145 * If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C' : 00146 * If INFO .EQ. 0 : 00147 * RANKA orthonormal columns of U are returned in the 00148 * leading RANKA columns of the array A. Here RANKA <= N 00149 * is the number of computed singular values of A that are 00150 * above the underflow threshold DLAMCH('S'). The singular 00151 * vectors corresponding to underflowed or zero singular 00152 * values are not computed. The value of RANKA is returned 00153 * in the array WORK as RANKA=NINT(WORK(2)). Also see the 00154 * descriptions of SVA and WORK. The computed columns of U 00155 * are mutually numerically orthogonal up to approximately 00156 * TOL=DSQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'), 00157 * see the description of JOBU. 00158 * If INFO .GT. 0 : 00159 * the procedure DGESVJ did not converge in the given number 00160 * of iterations (sweeps). In that case, the computed 00161 * columns of U may not be orthogonal up to TOL. The output 00162 * U (stored in A), SIGMA (given by the computed singular 00163 * values in SVA(1:N)) and V is still a decomposition of the 00164 * input matrix A in the sense that the residual 00165 * ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small. 00166 * 00167 * If JOBU .EQ. 'N' : 00168 * If INFO .EQ. 0 : 00169 * Note that the left singular vectors are 'for free' in the 00170 * one-sided Jacobi SVD algorithm. However, if only the 00171 * singular values are needed, the level of numerical 00172 * orthogonality of U is not an issue and iterations are 00173 * stopped when the columns of the iterated matrix are 00174 * numerically orthogonal up to approximately M*EPS. Thus, 00175 * on exit, A contains the columns of U scaled with the 00176 * corresponding singular values. 00177 * If INFO .GT. 0 : 00178 * the procedure DGESVJ did not converge in the given number 00179 * of iterations (sweeps). 00180 * 00181 * LDA (input) INTEGER 00182 * The leading dimension of the array A. LDA >= max(1,M). 00183 * 00184 * SVA (workspace/output) DOUBLE PRECISION array, dimension (N) 00185 * On exit : 00186 * If INFO .EQ. 0 : 00187 * depending on the value SCALE = WORK(1), we have: 00188 * If SCALE .EQ. ONE : 00189 * SVA(1:N) contains the computed singular values of A. 00190 * During the computation SVA contains the Euclidean column 00191 * norms of the iterated matrices in the array A. 00192 * If SCALE .NE. ONE : 00193 * The singular values of A are SCALE*SVA(1:N), and this 00194 * factored representation is due to the fact that some of the 00195 * singular values of A might underflow or overflow. 00196 * If INFO .GT. 0 : 00197 * the procedure DGESVJ did not converge in the given number of 00198 * iterations (sweeps) and SCALE*SVA(1:N) may not be accurate. 00199 * 00200 * MV (input) INTEGER 00201 * If JOBV .EQ. 'A', then the product of Jacobi rotations in DGESVJ 00202 * is applied to the first MV rows of V. See the description of JOBV. 00203 * 00204 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N) 00205 * If JOBV = 'V', then V contains on exit the N-by-N matrix of 00206 * the right singular vectors; 00207 * If JOBV = 'A', then V contains the product of the computed right 00208 * singular vector matrix and the initial matrix in 00209 * the array V. 00210 * If JOBV = 'N', then V is not referenced. 00211 * 00212 * LDV (input) INTEGER 00213 * The leading dimension of the array V, LDV .GE. 1. 00214 * If JOBV .EQ. 'V', then LDV .GE. max(1,N). 00215 * If JOBV .EQ. 'A', then LDV .GE. max(1,MV) . 00216 * 00217 * WORK (input/workspace/output) DOUBLE PRECISION array, dimension max(4,M+N). 00218 * On entry : 00219 * If JOBU .EQ. 'C' : 00220 * WORK(1) = CTOL, where CTOL defines the threshold for convergence. 00221 * The process stops if all columns of A are mutually 00222 * orthogonal up to CTOL*EPS, EPS=DLAMCH('E'). 00223 * It is required that CTOL >= ONE, i.e. it is not 00224 * allowed to force the routine to obtain orthogonality 00225 * below EPS. 00226 * On exit : 00227 * WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N) 00228 * are the computed singular values of A. 00229 * (See description of SVA().) 00230 * WORK(2) = NINT(WORK(2)) is the number of the computed nonzero 00231 * singular values. 00232 * WORK(3) = NINT(WORK(3)) is the number of the computed singular 00233 * values that are larger than the underflow threshold. 00234 * WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi 00235 * rotations needed for numerical convergence. 00236 * WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep. 00237 * This is useful information in cases when DGESVJ did 00238 * not converge, as it can be used to estimate whether 00239 * the output is stil useful and for post festum analysis. 00240 * WORK(6) = the largest absolute value over all sines of the 00241 * Jacobi rotation angles in the last sweep. It can be 00242 * useful for a post festum analysis. 00243 * 00244 * LWORK (input) INTEGER 00245 * length of WORK, WORK >= MAX(6,M+N) 00246 * 00247 * INFO (output) INTEGER 00248 * = 0 : successful exit. 00249 * < 0 : if INFO = -i, then the i-th argument had an illegal value 00250 * > 0 : DGESVJ did not converge in the maximal allowed number (30) 00251 * of sweeps. The output may still be useful. See the 00252 * description of WORK. 00253 * 00254 * ===================================================================== 00255 * 00256 * .. Local Parameters .. 00257 DOUBLE PRECISION ZERO, HALF, ONE, TWO 00258 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0, 00259 + TWO = 2.0D0 ) 00260 INTEGER NSWEEP 00261 PARAMETER ( NSWEEP = 30 ) 00262 * .. 00263 * .. Local Scalars .. 00264 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG, 00265 + BIGTHETA, CS, CTOL, EPSLN, LARGE, MXAAPQ, 00266 + MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL, 00267 + SKL, SFMIN, SMALL, SN, T, TEMP1, THETA, 00268 + THSIGN, TOL 00269 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1, 00270 + ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34, 00271 + N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP, 00272 + SWBAND 00273 LOGICAL APPLV, GOSCALE, LOWER, LSVEC, NOSCALE, ROTOK, 00274 + RSVEC, UCTOL, UPPER 00275 * .. 00276 * .. Local Arrays .. 00277 DOUBLE PRECISION FASTR( 5 ) 00278 * .. 00279 * .. Intrinsic Functions .. 00280 INTRINSIC DABS, DMAX1, DMIN1, DBLE, MIN0, DSIGN, DSQRT 00281 * .. 00282 * .. External Functions .. 00283 * .. 00284 * from BLAS 00285 DOUBLE PRECISION DDOT, DNRM2 00286 EXTERNAL DDOT, DNRM2 00287 INTEGER IDAMAX 00288 EXTERNAL IDAMAX 00289 * from LAPACK 00290 DOUBLE PRECISION DLAMCH 00291 EXTERNAL DLAMCH 00292 LOGICAL LSAME 00293 EXTERNAL LSAME 00294 * .. 00295 * .. External Subroutines .. 00296 * .. 00297 * from BLAS 00298 EXTERNAL DAXPY, DCOPY, DROTM, DSCAL, DSWAP 00299 * from LAPACK 00300 EXTERNAL DLASCL, DLASET, DLASSQ, XERBLA 00301 * 00302 EXTERNAL DGSVJ0, DGSVJ1 00303 * .. 00304 * .. Executable Statements .. 00305 * 00306 * Test the input arguments 00307 * 00308 LSVEC = LSAME( JOBU, 'U' ) 00309 UCTOL = LSAME( JOBU, 'C' ) 00310 RSVEC = LSAME( JOBV, 'V' ) 00311 APPLV = LSAME( JOBV, 'A' ) 00312 UPPER = LSAME( JOBA, 'U' ) 00313 LOWER = LSAME( JOBA, 'L' ) 00314 * 00315 IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN 00316 INFO = -1 00317 ELSE IF( .NOT.( LSVEC .OR. UCTOL .OR. LSAME( JOBU, 'N' ) ) ) THEN 00318 INFO = -2 00319 ELSE IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN 00320 INFO = -3 00321 ELSE IF( M.LT.0 ) THEN 00322 INFO = -4 00323 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN 00324 INFO = -5 00325 ELSE IF( LDA.LT.M ) THEN 00326 INFO = -7 00327 ELSE IF( MV.LT.0 ) THEN 00328 INFO = -9 00329 ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR. 00330 + ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN 00331 INFO = -11 00332 ELSE IF( UCTOL .AND. ( WORK( 1 ).LE.ONE ) ) THEN 00333 INFO = -12 00334 ELSE IF( LWORK.LT.MAX0( M+N, 6 ) ) THEN 00335 INFO = -13 00336 ELSE 00337 INFO = 0 00338 END IF 00339 * 00340 * #:( 00341 IF( INFO.NE.0 ) THEN 00342 CALL XERBLA( 'DGESVJ', -INFO ) 00343 RETURN 00344 END IF 00345 * 00346 * #:) Quick return for void matrix 00347 * 00348 IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )RETURN 00349 * 00350 * Set numerical parameters 00351 * The stopping criterion for Jacobi rotations is 00352 * 00353 * max_{i<>j}|A(:,i)^T * A(:,j)|/(||A(:,i)||*||A(:,j)||) < CTOL*EPS 00354 * 00355 * where EPS is the round-off and CTOL is defined as follows: 00356 * 00357 IF( UCTOL ) THEN 00358 * ... user controlled 00359 CTOL = WORK( 1 ) 00360 ELSE 00361 * ... default 00362 IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN 00363 CTOL = DSQRT( DBLE( M ) ) 00364 ELSE 00365 CTOL = DBLE( M ) 00366 END IF 00367 END IF 00368 * ... and the machine dependent parameters are 00369 *[!] (Make sure that DLAMCH() works properly on the target machine.) 00370 * 00371 EPSLN = DLAMCH( 'Epsilon' ) 00372 ROOTEPS = DSQRT( EPSLN ) 00373 SFMIN = DLAMCH( 'SafeMinimum' ) 00374 ROOTSFMIN = DSQRT( SFMIN ) 00375 SMALL = SFMIN / EPSLN 00376 BIG = DLAMCH( 'Overflow' ) 00377 * BIG = ONE / SFMIN 00378 ROOTBIG = ONE / ROOTSFMIN 00379 LARGE = BIG / DSQRT( DBLE( M*N ) ) 00380 BIGTHETA = ONE / ROOTEPS 00381 * 00382 TOL = CTOL*EPSLN 00383 ROOTTOL = DSQRT( TOL ) 00384 * 00385 IF( DBLE( M )*EPSLN.GE.ONE ) THEN 00386 INFO = -5 00387 CALL XERBLA( 'DGESVJ', -INFO ) 00388 RETURN 00389 END IF 00390 * 00391 * Initialize the right singular vector matrix. 00392 * 00393 IF( RSVEC ) THEN 00394 MVL = N 00395 CALL DLASET( 'A', MVL, N, ZERO, ONE, V, LDV ) 00396 ELSE IF( APPLV ) THEN 00397 MVL = MV 00398 END IF 00399 RSVEC = RSVEC .OR. APPLV 00400 * 00401 * Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N ) 00402 *(!) If necessary, scale A to protect the largest singular value 00403 * from overflow. It is possible that saving the largest singular 00404 * value destroys the information about the small ones. 00405 * This initial scaling is almost minimal in the sense that the 00406 * goal is to make sure that no column norm overflows, and that 00407 * DSQRT(N)*max_i SVA(i) does not overflow. If INFinite entries 00408 * in A are detected, the procedure returns with INFO=-6. 00409 * 00410 SKL= ONE / DSQRT( DBLE( M )*DBLE( N ) ) 00411 NOSCALE = .TRUE. 00412 GOSCALE = .TRUE. 00413 * 00414 IF( LOWER ) THEN 00415 * the input matrix is M-by-N lower triangular (trapezoidal) 00416 DO 1874 p = 1, N 00417 AAPP = ZERO 00418 AAQQ = ONE 00419 CALL DLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ ) 00420 IF( AAPP.GT.BIG ) THEN 00421 INFO = -6 00422 CALL XERBLA( 'DGESVJ', -INFO ) 00423 RETURN 00424 END IF 00425 AAQQ = DSQRT( AAQQ ) 00426 IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN 00427 SVA( p ) = AAPP*AAQQ 00428 ELSE 00429 NOSCALE = .FALSE. 00430 SVA( p ) = AAPP*( AAQQ*SKL) 00431 IF( GOSCALE ) THEN 00432 GOSCALE = .FALSE. 00433 DO 1873 q = 1, p - 1 00434 SVA( q ) = SVA( q )*SKL 00435 1873 CONTINUE 00436 END IF 00437 END IF 00438 1874 CONTINUE 00439 ELSE IF( UPPER ) THEN 00440 * the input matrix is M-by-N upper triangular (trapezoidal) 00441 DO 2874 p = 1, N 00442 AAPP = ZERO 00443 AAQQ = ONE 00444 CALL DLASSQ( p, A( 1, p ), 1, AAPP, AAQQ ) 00445 IF( AAPP.GT.BIG ) THEN 00446 INFO = -6 00447 CALL XERBLA( 'DGESVJ', -INFO ) 00448 RETURN 00449 END IF 00450 AAQQ = DSQRT( AAQQ ) 00451 IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN 00452 SVA( p ) = AAPP*AAQQ 00453 ELSE 00454 NOSCALE = .FALSE. 00455 SVA( p ) = AAPP*( AAQQ*SKL) 00456 IF( GOSCALE ) THEN 00457 GOSCALE = .FALSE. 00458 DO 2873 q = 1, p - 1 00459 SVA( q ) = SVA( q )*SKL 00460 2873 CONTINUE 00461 END IF 00462 END IF 00463 2874 CONTINUE 00464 ELSE 00465 * the input matrix is M-by-N general dense 00466 DO 3874 p = 1, N 00467 AAPP = ZERO 00468 AAQQ = ONE 00469 CALL DLASSQ( M, A( 1, p ), 1, AAPP, AAQQ ) 00470 IF( AAPP.GT.BIG ) THEN 00471 INFO = -6 00472 CALL XERBLA( 'DGESVJ', -INFO ) 00473 RETURN 00474 END IF 00475 AAQQ = DSQRT( AAQQ ) 00476 IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN 00477 SVA( p ) = AAPP*AAQQ 00478 ELSE 00479 NOSCALE = .FALSE. 00480 SVA( p ) = AAPP*( AAQQ*SKL) 00481 IF( GOSCALE ) THEN 00482 GOSCALE = .FALSE. 00483 DO 3873 q = 1, p - 1 00484 SVA( q ) = SVA( q )*SKL 00485 3873 CONTINUE 00486 END IF 00487 END IF 00488 3874 CONTINUE 00489 END IF 00490 * 00491 IF( NOSCALE )SKL= ONE 00492 * 00493 * Move the smaller part of the spectrum from the underflow threshold 00494 *(!) Start by determining the position of the nonzero entries of the 00495 * array SVA() relative to ( SFMIN, BIG ). 00496 * 00497 AAPP = ZERO 00498 AAQQ = BIG 00499 DO 4781 p = 1, N 00500 IF( SVA( p ).NE.ZERO )AAQQ = DMIN1( AAQQ, SVA( p ) ) 00501 AAPP = DMAX1( AAPP, SVA( p ) ) 00502 4781 CONTINUE 00503 * 00504 * #:) Quick return for zero matrix 00505 * 00506 IF( AAPP.EQ.ZERO ) THEN 00507 IF( LSVEC )CALL DLASET( 'G', M, N, ZERO, ONE, A, LDA ) 00508 WORK( 1 ) = ONE 00509 WORK( 2 ) = ZERO 00510 WORK( 3 ) = ZERO 00511 WORK( 4 ) = ZERO 00512 WORK( 5 ) = ZERO 00513 WORK( 6 ) = ZERO 00514 RETURN 00515 END IF 00516 * 00517 * #:) Quick return for one-column matrix 00518 * 00519 IF( N.EQ.1 ) THEN 00520 IF( LSVEC )CALL DLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1, 00521 + A( 1, 1 ), LDA, IERR ) 00522 WORK( 1 ) = ONE / SKL 00523 IF( SVA( 1 ).GE.SFMIN ) THEN 00524 WORK( 2 ) = ONE 00525 ELSE 00526 WORK( 2 ) = ZERO 00527 END IF 00528 WORK( 3 ) = ZERO 00529 WORK( 4 ) = ZERO 00530 WORK( 5 ) = ZERO 00531 WORK( 6 ) = ZERO 00532 RETURN 00533 END IF 00534 * 00535 * Protect small singular values from underflow, and try to 00536 * avoid underflows/overflows in computing Jacobi rotations. 00537 * 00538 SN = DSQRT( SFMIN / EPSLN ) 00539 TEMP1 = DSQRT( BIG / DBLE( N ) ) 00540 IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR. 00541 + ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN 00542 TEMP1 = DMIN1( BIG, TEMP1 / AAPP ) 00543 * AAQQ = AAQQ*TEMP1 00544 * AAPP = AAPP*TEMP1 00545 ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN 00546 TEMP1 = DMIN1( SN / AAQQ, BIG / ( AAPP*DSQRT( DBLE( N ) ) ) ) 00547 * AAQQ = AAQQ*TEMP1 00548 * AAPP = AAPP*TEMP1 00549 ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN 00550 TEMP1 = DMAX1( SN / AAQQ, TEMP1 / AAPP ) 00551 * AAQQ = AAQQ*TEMP1 00552 * AAPP = AAPP*TEMP1 00553 ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN 00554 TEMP1 = DMIN1( SN / AAQQ, BIG / ( DSQRT( DBLE( N ) )*AAPP ) ) 00555 * AAQQ = AAQQ*TEMP1 00556 * AAPP = AAPP*TEMP1 00557 ELSE 00558 TEMP1 = ONE 00559 END IF 00560 * 00561 * Scale, if necessary 00562 * 00563 IF( TEMP1.NE.ONE ) THEN 00564 CALL DLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR ) 00565 END IF 00566 SKL= TEMP1*SKL 00567 IF( SKL.NE.ONE ) THEN 00568 CALL DLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR ) 00569 SKL= ONE / SKL 00570 END IF 00571 * 00572 * Row-cyclic Jacobi SVD algorithm with column pivoting 00573 * 00574 EMPTSW = ( N*( N-1 ) ) / 2 00575 NOTROT = 0 00576 FASTR( 1 ) = ZERO 00577 * 00578 * A is represented in factored form A = A * diag(WORK), where diag(WORK) 00579 * is initialized to identity. WORK is updated during fast scaled 00580 * rotations. 00581 * 00582 DO 1868 q = 1, N 00583 WORK( q ) = ONE 00584 1868 CONTINUE 00585 * 00586 * 00587 SWBAND = 3 00588 *[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective 00589 * if DGESVJ is used as a computational routine in the preconditioned 00590 * Jacobi SVD algorithm DGESVJ. For sweeps i=1:SWBAND the procedure 00591 * works on pivots inside a band-like region around the diagonal. 00592 * The boundaries are determined dynamically, based on the number of 00593 * pivots above a threshold. 00594 * 00595 KBL = MIN0( 8, N ) 00596 *[TP] KBL is a tuning parameter that defines the tile size in the 00597 * tiling of the p-q loops of pivot pairs. In general, an optimal 00598 * value of KBL depends on the matrix dimensions and on the 00599 * parameters of the computer's memory. 00600 * 00601 NBL = N / KBL 00602 IF( ( NBL*KBL ).NE.N )NBL = NBL + 1 00603 * 00604 BLSKIP = KBL**2 00605 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. 00606 * 00607 ROWSKIP = MIN0( 5, KBL ) 00608 *[TP] ROWSKIP is a tuning parameter. 00609 * 00610 LKAHEAD = 1 00611 *[TP] LKAHEAD is a tuning parameter. 00612 * 00613 * Quasi block transformations, using the lower (upper) triangular 00614 * structure of the input matrix. The quasi-block-cycling usually 00615 * invokes cubic convergence. Big part of this cycle is done inside 00616 * canonical subspaces of dimensions less than M. 00617 * 00618 IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX0( 64, 4*KBL ) ) ) THEN 00619 *[TP] The number of partition levels and the actual partition are 00620 * tuning parameters. 00621 N4 = N / 4 00622 N2 = N / 2 00623 N34 = 3*N4 00624 IF( APPLV ) THEN 00625 q = 0 00626 ELSE 00627 q = 1 00628 END IF 00629 * 00630 IF( LOWER ) THEN 00631 * 00632 * This works very well on lower triangular matrices, in particular 00633 * in the framework of the preconditioned Jacobi SVD (xGEJSV). 00634 * The idea is simple: 00635 * [+ 0 0 0] Note that Jacobi transformations of [0 0] 00636 * [+ + 0 0] [0 0] 00637 * [+ + x 0] actually work on [x 0] [x 0] 00638 * [+ + x x] [x x]. [x x] 00639 * 00640 CALL DGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA, 00641 + WORK( N34+1 ), SVA( N34+1 ), MVL, 00642 + V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL, 00643 + 2, WORK( N+1 ), LWORK-N, IERR ) 00644 * 00645 CALL DGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA, 00646 + WORK( N2+1 ), SVA( N2+1 ), MVL, 00647 + V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2, 00648 + WORK( N+1 ), LWORK-N, IERR ) 00649 * 00650 CALL DGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA, 00651 + WORK( N2+1 ), SVA( N2+1 ), MVL, 00652 + V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1, 00653 + WORK( N+1 ), LWORK-N, IERR ) 00654 * 00655 CALL DGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA, 00656 + WORK( N4+1 ), SVA( N4+1 ), MVL, 00657 + V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1, 00658 + WORK( N+1 ), LWORK-N, IERR ) 00659 * 00660 CALL DGSVJ0( JOBV, M, N4, A, LDA, WORK, SVA, MVL, V, LDV, 00661 + EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N, 00662 + IERR ) 00663 * 00664 CALL DGSVJ1( JOBV, M, N2, N4, A, LDA, WORK, SVA, MVL, V, 00665 + LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ), 00666 + LWORK-N, IERR ) 00667 * 00668 * 00669 ELSE IF( UPPER ) THEN 00670 * 00671 * 00672 CALL DGSVJ0( JOBV, N4, N4, A, LDA, WORK, SVA, MVL, V, LDV, 00673 + EPSLN, SFMIN, TOL, 2, WORK( N+1 ), LWORK-N, 00674 + IERR ) 00675 * 00676 CALL DGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA, WORK( N4+1 ), 00677 + SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV, 00678 + EPSLN, SFMIN, TOL, 1, WORK( N+1 ), LWORK-N, 00679 + IERR ) 00680 * 00681 CALL DGSVJ1( JOBV, N2, N2, N4, A, LDA, WORK, SVA, MVL, V, 00682 + LDV, EPSLN, SFMIN, TOL, 1, WORK( N+1 ), 00683 + LWORK-N, IERR ) 00684 * 00685 CALL DGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA, 00686 + WORK( N2+1 ), SVA( N2+1 ), MVL, 00687 + V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1, 00688 + WORK( N+1 ), LWORK-N, IERR ) 00689 00690 END IF 00691 * 00692 END IF 00693 * 00694 * .. Row-cyclic pivot strategy with de Rijk's pivoting .. 00695 * 00696 DO 1993 i = 1, NSWEEP 00697 * 00698 * .. go go go ... 00699 * 00700 MXAAPQ = ZERO 00701 MXSINJ = ZERO 00702 ISWROT = 0 00703 * 00704 NOTROT = 0 00705 PSKIPPED = 0 00706 * 00707 * Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs 00708 * 1 <= p < q <= N. This is the first step toward a blocked implementation 00709 * of the rotations. New implementation, based on block transformations, 00710 * is under development. 00711 * 00712 DO 2000 ibr = 1, NBL 00713 * 00714 igl = ( ibr-1 )*KBL + 1 00715 * 00716 DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr ) 00717 * 00718 igl = igl + ir1*KBL 00719 * 00720 DO 2001 p = igl, MIN0( igl+KBL-1, N-1 ) 00721 * 00722 * .. de Rijk's pivoting 00723 * 00724 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 00725 IF( p.NE.q ) THEN 00726 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 00727 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, 00728 + V( 1, q ), 1 ) 00729 TEMP1 = SVA( p ) 00730 SVA( p ) = SVA( q ) 00731 SVA( q ) = TEMP1 00732 TEMP1 = WORK( p ) 00733 WORK( p ) = WORK( q ) 00734 WORK( q ) = TEMP1 00735 END IF 00736 * 00737 IF( ir1.EQ.0 ) THEN 00738 * 00739 * Column norms are periodically updated by explicit 00740 * norm computation. 00741 * Caveat: 00742 * Unfortunately, some BLAS implementations compute DNRM2(M,A(1,p),1) 00743 * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may cause the result to 00744 * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and to 00745 * underflow for ||A(:,p)||_2 < DSQRT(underflow_threshold). 00746 * Hence, DNRM2 cannot be trusted, not even in the case when 00747 * the true norm is far from the under(over)flow boundaries. 00748 * If properly implemented DNRM2 is available, the IF-THEN-ELSE 00749 * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * WORK(p)". 00750 * 00751 IF( ( SVA( p ).LT.ROOTBIG ) .AND. 00752 + ( SVA( p ).GT.ROOTSFMIN ) ) THEN 00753 SVA( p ) = DNRM2( M, A( 1, p ), 1 )*WORK( p ) 00754 ELSE 00755 TEMP1 = ZERO 00756 AAPP = ONE 00757 CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP ) 00758 SVA( p ) = TEMP1*DSQRT( AAPP )*WORK( p ) 00759 END IF 00760 AAPP = SVA( p ) 00761 ELSE 00762 AAPP = SVA( p ) 00763 END IF 00764 * 00765 IF( AAPP.GT.ZERO ) THEN 00766 * 00767 PSKIPPED = 0 00768 * 00769 DO 2002 q = p + 1, MIN0( igl+KBL-1, N ) 00770 * 00771 AAQQ = SVA( q ) 00772 * 00773 IF( AAQQ.GT.ZERO ) THEN 00774 * 00775 AAPP0 = AAPP 00776 IF( AAQQ.GE.ONE ) THEN 00777 ROTOK = ( SMALL*AAPP ).LE.AAQQ 00778 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 00779 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00780 + q ), 1 )*WORK( p )*WORK( q ) / 00781 + AAQQ ) / AAPP 00782 ELSE 00783 CALL DCOPY( M, A( 1, p ), 1, 00784 + WORK( N+1 ), 1 ) 00785 CALL DLASCL( 'G', 0, 0, AAPP, 00786 + WORK( p ), M, 1, 00787 + WORK( N+1 ), LDA, IERR ) 00788 AAPQ = DDOT( M, WORK( N+1 ), 1, 00789 + A( 1, q ), 1 )*WORK( q ) / AAQQ 00790 END IF 00791 ELSE 00792 ROTOK = AAPP.LE.( AAQQ / SMALL ) 00793 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 00794 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00795 + q ), 1 )*WORK( p )*WORK( q ) / 00796 + AAQQ ) / AAPP 00797 ELSE 00798 CALL DCOPY( M, A( 1, q ), 1, 00799 + WORK( N+1 ), 1 ) 00800 CALL DLASCL( 'G', 0, 0, AAQQ, 00801 + WORK( q ), M, 1, 00802 + WORK( N+1 ), LDA, IERR ) 00803 AAPQ = DDOT( M, WORK( N+1 ), 1, 00804 + A( 1, p ), 1 )*WORK( p ) / AAPP 00805 END IF 00806 END IF 00807 * 00808 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) ) 00809 * 00810 * TO rotate or NOT to rotate, THAT is the question ... 00811 * 00812 IF( DABS( AAPQ ).GT.TOL ) THEN 00813 * 00814 * .. rotate 00815 *[RTD] ROTATED = ROTATED + ONE 00816 * 00817 IF( ir1.EQ.0 ) THEN 00818 NOTROT = 0 00819 PSKIPPED = 0 00820 ISWROT = ISWROT + 1 00821 END IF 00822 * 00823 IF( ROTOK ) THEN 00824 * 00825 AQOAP = AAQQ / AAPP 00826 APOAQ = AAPP / AAQQ 00827 THETA = -HALF*DABS( AQOAP-APOAQ ) / 00828 + AAPQ 00829 * 00830 IF( DABS( THETA ).GT.BIGTHETA ) THEN 00831 * 00832 T = HALF / THETA 00833 FASTR( 3 ) = T*WORK( p ) / WORK( q ) 00834 FASTR( 4 ) = -T*WORK( q ) / 00835 + WORK( p ) 00836 CALL DROTM( M, A( 1, p ), 1, 00837 + A( 1, q ), 1, FASTR ) 00838 IF( RSVEC )CALL DROTM( MVL, 00839 + V( 1, p ), 1, 00840 + V( 1, q ), 1, 00841 + FASTR ) 00842 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00843 + ONE+T*APOAQ*AAPQ ) ) 00844 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00845 + ONE-T*AQOAP*AAPQ ) ) 00846 MXSINJ = DMAX1( MXSINJ, DABS( T ) ) 00847 * 00848 ELSE 00849 * 00850 * .. choose correct signum for THETA and rotate 00851 * 00852 THSIGN = -DSIGN( ONE, AAPQ ) 00853 T = ONE / ( THETA+THSIGN* 00854 + DSQRT( ONE+THETA*THETA ) ) 00855 CS = DSQRT( ONE / ( ONE+T*T ) ) 00856 SN = T*CS 00857 * 00858 MXSINJ = DMAX1( MXSINJ, DABS( SN ) ) 00859 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00860 + ONE+T*APOAQ*AAPQ ) ) 00861 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00862 + ONE-T*AQOAP*AAPQ ) ) 00863 * 00864 APOAQ = WORK( p ) / WORK( q ) 00865 AQOAP = WORK( q ) / WORK( p ) 00866 IF( WORK( p ).GE.ONE ) THEN 00867 IF( WORK( q ).GE.ONE ) THEN 00868 FASTR( 3 ) = T*APOAQ 00869 FASTR( 4 ) = -T*AQOAP 00870 WORK( p ) = WORK( p )*CS 00871 WORK( q ) = WORK( q )*CS 00872 CALL DROTM( M, A( 1, p ), 1, 00873 + A( 1, q ), 1, 00874 + FASTR ) 00875 IF( RSVEC )CALL DROTM( MVL, 00876 + V( 1, p ), 1, V( 1, q ), 00877 + 1, FASTR ) 00878 ELSE 00879 CALL DAXPY( M, -T*AQOAP, 00880 + A( 1, q ), 1, 00881 + A( 1, p ), 1 ) 00882 CALL DAXPY( M, CS*SN*APOAQ, 00883 + A( 1, p ), 1, 00884 + A( 1, q ), 1 ) 00885 WORK( p ) = WORK( p )*CS 00886 WORK( q ) = WORK( q ) / CS 00887 IF( RSVEC ) THEN 00888 CALL DAXPY( MVL, -T*AQOAP, 00889 + V( 1, q ), 1, 00890 + V( 1, p ), 1 ) 00891 CALL DAXPY( MVL, 00892 + CS*SN*APOAQ, 00893 + V( 1, p ), 1, 00894 + V( 1, q ), 1 ) 00895 END IF 00896 END IF 00897 ELSE 00898 IF( WORK( q ).GE.ONE ) THEN 00899 CALL DAXPY( M, T*APOAQ, 00900 + A( 1, p ), 1, 00901 + A( 1, q ), 1 ) 00902 CALL DAXPY( M, -CS*SN*AQOAP, 00903 + A( 1, q ), 1, 00904 + A( 1, p ), 1 ) 00905 WORK( p ) = WORK( p ) / CS 00906 WORK( q ) = WORK( q )*CS 00907 IF( RSVEC ) THEN 00908 CALL DAXPY( MVL, T*APOAQ, 00909 + V( 1, p ), 1, 00910 + V( 1, q ), 1 ) 00911 CALL DAXPY( MVL, 00912 + -CS*SN*AQOAP, 00913 + V( 1, q ), 1, 00914 + V( 1, p ), 1 ) 00915 END IF 00916 ELSE 00917 IF( WORK( p ).GE.WORK( q ) ) 00918 + THEN 00919 CALL DAXPY( M, -T*AQOAP, 00920 + A( 1, q ), 1, 00921 + A( 1, p ), 1 ) 00922 CALL DAXPY( M, CS*SN*APOAQ, 00923 + A( 1, p ), 1, 00924 + A( 1, q ), 1 ) 00925 WORK( p ) = WORK( p )*CS 00926 WORK( q ) = WORK( q ) / CS 00927 IF( RSVEC ) THEN 00928 CALL DAXPY( MVL, 00929 + -T*AQOAP, 00930 + V( 1, q ), 1, 00931 + V( 1, p ), 1 ) 00932 CALL DAXPY( MVL, 00933 + CS*SN*APOAQ, 00934 + V( 1, p ), 1, 00935 + V( 1, q ), 1 ) 00936 END IF 00937 ELSE 00938 CALL DAXPY( M, T*APOAQ, 00939 + A( 1, p ), 1, 00940 + A( 1, q ), 1 ) 00941 CALL DAXPY( M, 00942 + -CS*SN*AQOAP, 00943 + A( 1, q ), 1, 00944 + A( 1, p ), 1 ) 00945 WORK( p ) = WORK( p ) / CS 00946 WORK( q ) = WORK( q )*CS 00947 IF( RSVEC ) THEN 00948 CALL DAXPY( MVL, 00949 + T*APOAQ, V( 1, p ), 00950 + 1, V( 1, q ), 1 ) 00951 CALL DAXPY( MVL, 00952 + -CS*SN*AQOAP, 00953 + V( 1, q ), 1, 00954 + V( 1, p ), 1 ) 00955 END IF 00956 END IF 00957 END IF 00958 END IF 00959 END IF 00960 * 00961 ELSE 00962 * .. have to use modified Gram-Schmidt like transformation 00963 CALL DCOPY( M, A( 1, p ), 1, 00964 + WORK( N+1 ), 1 ) 00965 CALL DLASCL( 'G', 0, 0, AAPP, ONE, M, 00966 + 1, WORK( N+1 ), LDA, 00967 + IERR ) 00968 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M, 00969 + 1, A( 1, q ), LDA, IERR ) 00970 TEMP1 = -AAPQ*WORK( p ) / WORK( q ) 00971 CALL DAXPY( M, TEMP1, WORK( N+1 ), 1, 00972 + A( 1, q ), 1 ) 00973 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M, 00974 + 1, A( 1, q ), LDA, IERR ) 00975 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00976 + ONE-AAPQ*AAPQ ) ) 00977 MXSINJ = DMAX1( MXSINJ, SFMIN ) 00978 END IF 00979 * END IF ROTOK THEN ... ELSE 00980 * 00981 * In the case of cancellation in updating SVA(q), SVA(p) 00982 * recompute SVA(q), SVA(p). 00983 * 00984 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) 00985 + THEN 00986 IF( ( AAQQ.LT.ROOTBIG ) .AND. 00987 + ( AAQQ.GT.ROOTSFMIN ) ) THEN 00988 SVA( q ) = DNRM2( M, A( 1, q ), 1 )* 00989 + WORK( q ) 00990 ELSE 00991 T = ZERO 00992 AAQQ = ONE 00993 CALL DLASSQ( M, A( 1, q ), 1, T, 00994 + AAQQ ) 00995 SVA( q ) = T*DSQRT( AAQQ )*WORK( q ) 00996 END IF 00997 END IF 00998 IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN 00999 IF( ( AAPP.LT.ROOTBIG ) .AND. 01000 + ( AAPP.GT.ROOTSFMIN ) ) THEN 01001 AAPP = DNRM2( M, A( 1, p ), 1 )* 01002 + WORK( p ) 01003 ELSE 01004 T = ZERO 01005 AAPP = ONE 01006 CALL DLASSQ( M, A( 1, p ), 1, T, 01007 + AAPP ) 01008 AAPP = T*DSQRT( AAPP )*WORK( p ) 01009 END IF 01010 SVA( p ) = AAPP 01011 END IF 01012 * 01013 ELSE 01014 * A(:,p) and A(:,q) already numerically orthogonal 01015 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 01016 *[RTD] SKIPPED = SKIPPED + 1 01017 PSKIPPED = PSKIPPED + 1 01018 END IF 01019 ELSE 01020 * A(:,q) is zero column 01021 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 01022 PSKIPPED = PSKIPPED + 1 01023 END IF 01024 * 01025 IF( ( i.LE.SWBAND ) .AND. 01026 + ( PSKIPPED.GT.ROWSKIP ) ) THEN 01027 IF( ir1.EQ.0 )AAPP = -AAPP 01028 NOTROT = 0 01029 GO TO 2103 01030 END IF 01031 * 01032 2002 CONTINUE 01033 * END q-LOOP 01034 * 01035 2103 CONTINUE 01036 * bailed out of q-loop 01037 * 01038 SVA( p ) = AAPP 01039 * 01040 ELSE 01041 SVA( p ) = AAPP 01042 IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) ) 01043 + NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p 01044 END IF 01045 * 01046 2001 CONTINUE 01047 * end of the p-loop 01048 * end of doing the block ( ibr, ibr ) 01049 1002 CONTINUE 01050 * end of ir1-loop 01051 * 01052 * ... go to the off diagonal blocks 01053 * 01054 igl = ( ibr-1 )*KBL + 1 01055 * 01056 DO 2010 jbc = ibr + 1, NBL 01057 * 01058 jgl = ( jbc-1 )*KBL + 1 01059 * 01060 * doing the block at ( ibr, jbc ) 01061 * 01062 IJBLSK = 0 01063 DO 2100 p = igl, MIN0( igl+KBL-1, N ) 01064 * 01065 AAPP = SVA( p ) 01066 IF( AAPP.GT.ZERO ) THEN 01067 * 01068 PSKIPPED = 0 01069 * 01070 DO 2200 q = jgl, MIN0( jgl+KBL-1, N ) 01071 * 01072 AAQQ = SVA( q ) 01073 IF( AAQQ.GT.ZERO ) THEN 01074 AAPP0 = AAPP 01075 * 01076 * .. M x 2 Jacobi SVD .. 01077 * 01078 * Safe Gram matrix computation 01079 * 01080 IF( AAQQ.GE.ONE ) THEN 01081 IF( AAPP.GE.AAQQ ) THEN 01082 ROTOK = ( SMALL*AAPP ).LE.AAQQ 01083 ELSE 01084 ROTOK = ( SMALL*AAQQ ).LE.AAPP 01085 END IF 01086 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 01087 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 01088 + q ), 1 )*WORK( p )*WORK( q ) / 01089 + AAQQ ) / AAPP 01090 ELSE 01091 CALL DCOPY( M, A( 1, p ), 1, 01092 + WORK( N+1 ), 1 ) 01093 CALL DLASCL( 'G', 0, 0, AAPP, 01094 + WORK( p ), M, 1, 01095 + WORK( N+1 ), LDA, IERR ) 01096 AAPQ = DDOT( M, WORK( N+1 ), 1, 01097 + A( 1, q ), 1 )*WORK( q ) / AAQQ 01098 END IF 01099 ELSE 01100 IF( AAPP.GE.AAQQ ) THEN 01101 ROTOK = AAPP.LE.( AAQQ / SMALL ) 01102 ELSE 01103 ROTOK = AAQQ.LE.( AAPP / SMALL ) 01104 END IF 01105 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 01106 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 01107 + q ), 1 )*WORK( p )*WORK( q ) / 01108 + AAQQ ) / AAPP 01109 ELSE 01110 CALL DCOPY( M, A( 1, q ), 1, 01111 + WORK( N+1 ), 1 ) 01112 CALL DLASCL( 'G', 0, 0, AAQQ, 01113 + WORK( q ), M, 1, 01114 + WORK( N+1 ), LDA, IERR ) 01115 AAPQ = DDOT( M, WORK( N+1 ), 1, 01116 + A( 1, p ), 1 )*WORK( p ) / AAPP 01117 END IF 01118 END IF 01119 * 01120 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) ) 01121 * 01122 * TO rotate or NOT to rotate, THAT is the question ... 01123 * 01124 IF( DABS( AAPQ ).GT.TOL ) THEN 01125 NOTROT = 0 01126 *[RTD] ROTATED = ROTATED + 1 01127 PSKIPPED = 0 01128 ISWROT = ISWROT + 1 01129 * 01130 IF( ROTOK ) THEN 01131 * 01132 AQOAP = AAQQ / AAPP 01133 APOAQ = AAPP / AAQQ 01134 THETA = -HALF*DABS( AQOAP-APOAQ ) / 01135 + AAPQ 01136 IF( AAQQ.GT.AAPP0 )THETA = -THETA 01137 * 01138 IF( DABS( THETA ).GT.BIGTHETA ) THEN 01139 T = HALF / THETA 01140 FASTR( 3 ) = T*WORK( p ) / WORK( q ) 01141 FASTR( 4 ) = -T*WORK( q ) / 01142 + WORK( p ) 01143 CALL DROTM( M, A( 1, p ), 1, 01144 + A( 1, q ), 1, FASTR ) 01145 IF( RSVEC )CALL DROTM( MVL, 01146 + V( 1, p ), 1, 01147 + V( 1, q ), 1, 01148 + FASTR ) 01149 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 01150 + ONE+T*APOAQ*AAPQ ) ) 01151 AAPP = AAPP*DSQRT( DMAX1( ZERO, 01152 + ONE-T*AQOAP*AAPQ ) ) 01153 MXSINJ = DMAX1( MXSINJ, DABS( T ) ) 01154 ELSE 01155 * 01156 * .. choose correct signum for THETA and rotate 01157 * 01158 THSIGN = -DSIGN( ONE, AAPQ ) 01159 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN 01160 T = ONE / ( THETA+THSIGN* 01161 + DSQRT( ONE+THETA*THETA ) ) 01162 CS = DSQRT( ONE / ( ONE+T*T ) ) 01163 SN = T*CS 01164 MXSINJ = DMAX1( MXSINJ, DABS( SN ) ) 01165 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 01166 + ONE+T*APOAQ*AAPQ ) ) 01167 AAPP = AAPP*DSQRT( DMAX1( ZERO, 01168 + ONE-T*AQOAP*AAPQ ) ) 01169 * 01170 APOAQ = WORK( p ) / WORK( q ) 01171 AQOAP = WORK( q ) / WORK( p ) 01172 IF( WORK( p ).GE.ONE ) THEN 01173 * 01174 IF( WORK( q ).GE.ONE ) THEN 01175 FASTR( 3 ) = T*APOAQ 01176 FASTR( 4 ) = -T*AQOAP 01177 WORK( p ) = WORK( p )*CS 01178 WORK( q ) = WORK( q )*CS 01179 CALL DROTM( M, A( 1, p ), 1, 01180 + A( 1, q ), 1, 01181 + FASTR ) 01182 IF( RSVEC )CALL DROTM( MVL, 01183 + V( 1, p ), 1, V( 1, q ), 01184 + 1, FASTR ) 01185 ELSE 01186 CALL DAXPY( M, -T*AQOAP, 01187 + A( 1, q ), 1, 01188 + A( 1, p ), 1 ) 01189 CALL DAXPY( M, CS*SN*APOAQ, 01190 + A( 1, p ), 1, 01191 + A( 1, q ), 1 ) 01192 IF( RSVEC ) THEN 01193 CALL DAXPY( MVL, -T*AQOAP, 01194 + V( 1, q ), 1, 01195 + V( 1, p ), 1 ) 01196 CALL DAXPY( MVL, 01197 + CS*SN*APOAQ, 01198 + V( 1, p ), 1, 01199 + V( 1, q ), 1 ) 01200 END IF 01201 WORK( p ) = WORK( p )*CS 01202 WORK( q ) = WORK( q ) / CS 01203 END IF 01204 ELSE 01205 IF( WORK( q ).GE.ONE ) THEN 01206 CALL DAXPY( M, T*APOAQ, 01207 + A( 1, p ), 1, 01208 + A( 1, q ), 1 ) 01209 CALL DAXPY( M, -CS*SN*AQOAP, 01210 + A( 1, q ), 1, 01211 + A( 1, p ), 1 ) 01212 IF( RSVEC ) THEN 01213 CALL DAXPY( MVL, T*APOAQ, 01214 + V( 1, p ), 1, 01215 + V( 1, q ), 1 ) 01216 CALL DAXPY( MVL, 01217 + -CS*SN*AQOAP, 01218 + V( 1, q ), 1, 01219 + V( 1, p ), 1 ) 01220 END IF 01221 WORK( p ) = WORK( p ) / CS 01222 WORK( q ) = WORK( q )*CS 01223 ELSE 01224 IF( WORK( p ).GE.WORK( q ) ) 01225 + THEN 01226 CALL DAXPY( M, -T*AQOAP, 01227 + A( 1, q ), 1, 01228 + A( 1, p ), 1 ) 01229 CALL DAXPY( M, CS*SN*APOAQ, 01230 + A( 1, p ), 1, 01231 + A( 1, q ), 1 ) 01232 WORK( p ) = WORK( p )*CS 01233 WORK( q ) = WORK( q ) / CS 01234 IF( RSVEC ) THEN 01235 CALL DAXPY( MVL, 01236 + -T*AQOAP, 01237 + V( 1, q ), 1, 01238 + V( 1, p ), 1 ) 01239 CALL DAXPY( MVL, 01240 + CS*SN*APOAQ, 01241 + V( 1, p ), 1, 01242 + V( 1, q ), 1 ) 01243 END IF 01244 ELSE 01245 CALL DAXPY( M, T*APOAQ, 01246 + A( 1, p ), 1, 01247 + A( 1, q ), 1 ) 01248 CALL DAXPY( M, 01249 + -CS*SN*AQOAP, 01250 + A( 1, q ), 1, 01251 + A( 1, p ), 1 ) 01252 WORK( p ) = WORK( p ) / CS 01253 WORK( q ) = WORK( q )*CS 01254 IF( RSVEC ) THEN 01255 CALL DAXPY( MVL, 01256 + T*APOAQ, V( 1, p ), 01257 + 1, V( 1, q ), 1 ) 01258 CALL DAXPY( MVL, 01259 + -CS*SN*AQOAP, 01260 + V( 1, q ), 1, 01261 + V( 1, p ), 1 ) 01262 END IF 01263 END IF 01264 END IF 01265 END IF 01266 END IF 01267 * 01268 ELSE 01269 IF( AAPP.GT.AAQQ ) THEN 01270 CALL DCOPY( M, A( 1, p ), 1, 01271 + WORK( N+1 ), 1 ) 01272 CALL DLASCL( 'G', 0, 0, AAPP, ONE, 01273 + M, 1, WORK( N+1 ), LDA, 01274 + IERR ) 01275 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, 01276 + M, 1, A( 1, q ), LDA, 01277 + IERR ) 01278 TEMP1 = -AAPQ*WORK( p ) / WORK( q ) 01279 CALL DAXPY( M, TEMP1, WORK( N+1 ), 01280 + 1, A( 1, q ), 1 ) 01281 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, 01282 + M, 1, A( 1, q ), LDA, 01283 + IERR ) 01284 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 01285 + ONE-AAPQ*AAPQ ) ) 01286 MXSINJ = DMAX1( MXSINJ, SFMIN ) 01287 ELSE 01288 CALL DCOPY( M, A( 1, q ), 1, 01289 + WORK( N+1 ), 1 ) 01290 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, 01291 + M, 1, WORK( N+1 ), LDA, 01292 + IERR ) 01293 CALL DLASCL( 'G', 0, 0, AAPP, ONE, 01294 + M, 1, A( 1, p ), LDA, 01295 + IERR ) 01296 TEMP1 = -AAPQ*WORK( q ) / WORK( p ) 01297 CALL DAXPY( M, TEMP1, WORK( N+1 ), 01298 + 1, A( 1, p ), 1 ) 01299 CALL DLASCL( 'G', 0, 0, ONE, AAPP, 01300 + M, 1, A( 1, p ), LDA, 01301 + IERR ) 01302 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO, 01303 + ONE-AAPQ*AAPQ ) ) 01304 MXSINJ = DMAX1( MXSINJ, SFMIN ) 01305 END IF 01306 END IF 01307 * END IF ROTOK THEN ... ELSE 01308 * 01309 * In the case of cancellation in updating SVA(q) 01310 * .. recompute SVA(q) 01311 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) 01312 + THEN 01313 IF( ( AAQQ.LT.ROOTBIG ) .AND. 01314 + ( AAQQ.GT.ROOTSFMIN ) ) THEN 01315 SVA( q ) = DNRM2( M, A( 1, q ), 1 )* 01316 + WORK( q ) 01317 ELSE 01318 T = ZERO 01319 AAQQ = ONE 01320 CALL DLASSQ( M, A( 1, q ), 1, T, 01321 + AAQQ ) 01322 SVA( q ) = T*DSQRT( AAQQ )*WORK( q ) 01323 END IF 01324 END IF 01325 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN 01326 IF( ( AAPP.LT.ROOTBIG ) .AND. 01327 + ( AAPP.GT.ROOTSFMIN ) ) THEN 01328 AAPP = DNRM2( M, A( 1, p ), 1 )* 01329 + WORK( p ) 01330 ELSE 01331 T = ZERO 01332 AAPP = ONE 01333 CALL DLASSQ( M, A( 1, p ), 1, T, 01334 + AAPP ) 01335 AAPP = T*DSQRT( AAPP )*WORK( p ) 01336 END IF 01337 SVA( p ) = AAPP 01338 END IF 01339 * end of OK rotation 01340 ELSE 01341 NOTROT = NOTROT + 1 01342 *[RTD] SKIPPED = SKIPPED + 1 01343 PSKIPPED = PSKIPPED + 1 01344 IJBLSK = IJBLSK + 1 01345 END IF 01346 ELSE 01347 NOTROT = NOTROT + 1 01348 PSKIPPED = PSKIPPED + 1 01349 IJBLSK = IJBLSK + 1 01350 END IF 01351 * 01352 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) ) 01353 + THEN 01354 SVA( p ) = AAPP 01355 NOTROT = 0 01356 GO TO 2011 01357 END IF 01358 IF( ( i.LE.SWBAND ) .AND. 01359 + ( PSKIPPED.GT.ROWSKIP ) ) THEN 01360 AAPP = -AAPP 01361 NOTROT = 0 01362 GO TO 2203 01363 END IF 01364 * 01365 2200 CONTINUE 01366 * end of the q-loop 01367 2203 CONTINUE 01368 * 01369 SVA( p ) = AAPP 01370 * 01371 ELSE 01372 * 01373 IF( AAPP.EQ.ZERO )NOTROT = NOTROT + 01374 + MIN0( jgl+KBL-1, N ) - jgl + 1 01375 IF( AAPP.LT.ZERO )NOTROT = 0 01376 * 01377 END IF 01378 * 01379 2100 CONTINUE 01380 * end of the p-loop 01381 2010 CONTINUE 01382 * end of the jbc-loop 01383 2011 CONTINUE 01384 *2011 bailed out of the jbc-loop 01385 DO 2012 p = igl, MIN0( igl+KBL-1, N ) 01386 SVA( p ) = DABS( SVA( p ) ) 01387 2012 CONTINUE 01388 *** 01389 2000 CONTINUE 01390 *2000 :: end of the ibr-loop 01391 * 01392 * .. update SVA(N) 01393 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) ) 01394 + THEN 01395 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*WORK( N ) 01396 ELSE 01397 T = ZERO 01398 AAPP = ONE 01399 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP ) 01400 SVA( N ) = T*DSQRT( AAPP )*WORK( N ) 01401 END IF 01402 * 01403 * Additional steering devices 01404 * 01405 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR. 01406 + ( ISWROT.LE.N ) ) )SWBAND = i 01407 * 01408 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DSQRT( DBLE( N ) )* 01409 + TOL ) .AND. ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN 01410 GO TO 1994 01411 END IF 01412 * 01413 IF( NOTROT.GE.EMPTSW )GO TO 1994 01414 * 01415 1993 CONTINUE 01416 * end i=1:NSWEEP loop 01417 * 01418 * #:( Reaching this point means that the procedure has not converged. 01419 INFO = NSWEEP - 1 01420 GO TO 1995 01421 * 01422 1994 CONTINUE 01423 * #:) Reaching this point means numerical convergence after the i-th 01424 * sweep. 01425 * 01426 INFO = 0 01427 * #:) INFO = 0 confirms successful iterations. 01428 1995 CONTINUE 01429 * 01430 * Sort the singular values and find how many are above 01431 * the underflow threshold. 01432 * 01433 N2 = 0 01434 N4 = 0 01435 DO 5991 p = 1, N - 1 01436 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 01437 IF( p.NE.q ) THEN 01438 TEMP1 = SVA( p ) 01439 SVA( p ) = SVA( q ) 01440 SVA( q ) = TEMP1 01441 TEMP1 = WORK( p ) 01442 WORK( p ) = WORK( q ) 01443 WORK( q ) = TEMP1 01444 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 01445 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 ) 01446 END IF 01447 IF( SVA( p ).NE.ZERO ) THEN 01448 N4 = N4 + 1 01449 IF( SVA( p )*SKL.GT.SFMIN )N2 = N2 + 1 01450 END IF 01451 5991 CONTINUE 01452 IF( SVA( N ).NE.ZERO ) THEN 01453 N4 = N4 + 1 01454 IF( SVA( N )*SKL.GT.SFMIN )N2 = N2 + 1 01455 END IF 01456 * 01457 * Normalize the left singular vectors. 01458 * 01459 IF( LSVEC .OR. UCTOL ) THEN 01460 DO 1998 p = 1, N2 01461 CALL DSCAL( M, WORK( p ) / SVA( p ), A( 1, p ), 1 ) 01462 1998 CONTINUE 01463 END IF 01464 * 01465 * Scale the product of Jacobi rotations (assemble the fast rotations). 01466 * 01467 IF( RSVEC ) THEN 01468 IF( APPLV ) THEN 01469 DO 2398 p = 1, N 01470 CALL DSCAL( MVL, WORK( p ), V( 1, p ), 1 ) 01471 2398 CONTINUE 01472 ELSE 01473 DO 2399 p = 1, N 01474 TEMP1 = ONE / DNRM2( MVL, V( 1, p ), 1 ) 01475 CALL DSCAL( MVL, TEMP1, V( 1, p ), 1 ) 01476 2399 CONTINUE 01477 END IF 01478 END IF 01479 * 01480 * Undo scaling, if necessary (and possible). 01481 IF( ( ( SKL.GT.ONE ) .AND. ( SVA( 1 ).LT.( BIG / 01482 + SKL) ) ) .OR. ( ( SKL.LT.ONE ) .AND. ( SVA( N2 ).GT. 01483 + ( SFMIN / SKL) ) ) ) THEN 01484 DO 2400 p = 1, N 01485 SVA( p ) = SKL*SVA( p ) 01486 2400 CONTINUE 01487 SKL= ONE 01488 END IF 01489 * 01490 WORK( 1 ) = SKL 01491 * The singular values of A are SKL*SVA(1:N). If SKL.NE.ONE 01492 * then some of the singular values may overflow or underflow and 01493 * the spectrum is given in this factored representation. 01494 * 01495 WORK( 2 ) = DBLE( N4 ) 01496 * N4 is the number of computed nonzero singular values of A. 01497 * 01498 WORK( 3 ) = DBLE( N2 ) 01499 * N2 is the number of singular values of A greater than SFMIN. 01500 * If N2<N, SVA(N2:N) contains ZEROS and/or denormalized numbers 01501 * that may carry some information. 01502 * 01503 WORK( 4 ) = DBLE( i ) 01504 * i is the index of the last sweep before declaring convergence. 01505 * 01506 WORK( 5 ) = MXAAPQ 01507 * MXAAPQ is the largest absolute value of scaled pivots in the 01508 * last sweep 01509 * 01510 WORK( 6 ) = MXSINJ 01511 * MXSINJ is the largest absolute value of the sines of Jacobi angles 01512 * in the last sweep 01513 * 01514 RETURN 01515 * .. 01516 * .. END OF DGESVJ 01517 * .. 01518 END
1.7.3 
