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#include "rb_lapack.h"
extern VOID sgejsv_(char* joba, char* jobu, char* jobv, char* jobr, char* jobt, char* jobp, integer* m, integer* n, real* a, integer* lda, real* sva, real* u, integer* ldu, real* v, integer* ldv, real* work, integer* lwork, integer* iwork, integer* info);
static VALUE
rblapack_sgejsv(int argc, VALUE *argv, VALUE self){
VALUE rblapack_joba;
char joba;
VALUE rblapack_jobu;
char jobu;
VALUE rblapack_jobv;
char jobv;
VALUE rblapack_jobr;
char jobr;
VALUE rblapack_jobt;
char jobt;
VALUE rblapack_jobp;
char jobp;
VALUE rblapack_m;
integer m;
VALUE rblapack_a;
real *a;
VALUE rblapack_work;
real *work;
VALUE rblapack_lwork;
integer lwork;
VALUE rblapack_sva;
real *sva;
VALUE rblapack_u;
real *u;
VALUE rblapack_v;
real *v;
VALUE rblapack_iwork;
integer *iwork;
VALUE rblapack_info;
integer info;
VALUE rblapack_work_out__;
real *work_out__;
integer lda;
integer n;
integer ldu;
integer ldv;
VALUE rblapack_options;
if (argc > 0 && TYPE(argv[argc-1]) == T_HASH) {
argc--;
rblapack_options = argv[argc];
if (rb_hash_aref(rblapack_options, sHelp) == Qtrue) {
printf("%s\n", "USAGE:\n sva, u, v, iwork, info, work = NumRu::Lapack.sgejsv( joba, jobu, jobv, jobr, jobt, jobp, m, a, work, [:lwork => lwork, :usage => usage, :help => help])\n\n\nFORTRAN MANUAL\n SUBROUTINE SGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO )\n\n* Purpose\n* =======\n* SGEJSV computes the singular value decomposition (SVD) of a real M-by-N\n* matrix [A], where M >= N. The SVD of [A] is written as\n*\n* [A] = [U] * [SIGMA] * [V]^t,\n*\n* where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N\n* diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and\n* [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are\n* the singular values of [A]. The columns of [U] and [V] are the left and\n* the right singular vectors of [A], respectively. The matrices [U] and [V]\n* are computed and stored in the arrays U and V, respectively. The diagonal\n* of [SIGMA] is computed and stored in the array SVA.\n*\n\n* Arguments\n* =========\n*\n* JOBA (input) CHARACTER*1\n* Specifies the level of accuracy:\n* = 'C': This option works well (high relative accuracy) if A = B * D,\n* with well-conditioned B and arbitrary diagonal matrix D.\n* The accuracy cannot be spoiled by COLUMN scaling. The\n* accuracy of the computed output depends on the condition of\n* B, and the procedure aims at the best theoretical accuracy.\n* The relative error max_{i=1:N}|d sigma_i| / sigma_i is\n* bounded by f(M,N)*epsilon* cond(B), independent of D.\n* The input matrix is preprocessed with the QRF with column\n* pivoting. This initial preprocessing and preconditioning by\n* a rank revealing QR factorization is common for all values of\n* JOBA. Additional actions are specified as follows:\n* = 'E': Computation as with 'C' with an additional estimate of the\n* condition number of B. It provides a realistic error bound.\n* = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings\n* D1, D2, and well-conditioned matrix C, this option gives\n* higher accuracy than the 'C' option. If the structure of the\n* input matrix is not known, and relative accuracy is\n* desirable, then this option is advisable. The input matrix A\n* is preprocessed with QR factorization with FULL (row and\n* column) pivoting.\n* = 'G' Computation as with 'F' with an additional estimate of the\n* condition number of B, where A=D*B. If A has heavily weighted\n* rows, then using this condition number gives too pessimistic\n* error bound.\n* = 'A': Small singular values are the noise and the matrix is treated\n* as numerically rank defficient. The error in the computed\n* singular values is bounded by f(m,n)*epsilon*||A||.\n* The computed SVD A = U * S * V^t restores A up to\n* f(m,n)*epsilon*||A||.\n* This gives the procedure the licence to discard (set to zero)\n* all singular values below N*epsilon*||A||.\n* = 'R': Similar as in 'A'. Rank revealing property of the initial\n* QR factorization is used do reveal (using triangular factor)\n* a gap sigma_{r+1} < epsilon * sigma_r in which case the\n* numerical RANK is declared to be r. The SVD is computed with\n* absolute error bounds, but more accurately than with 'A'.\n* \n* JOBU (input) CHARACTER*1\n* Specifies whether to compute the columns of U:\n* = 'U': N columns of U are returned in the array U.\n* = 'F': full set of M left sing. vectors is returned in the array U.\n* = 'W': U may be used as workspace of length M*N. See the description\n* of U.\n* = 'N': U is not computed.\n* \n* JOBV (input) CHARACTER*1\n* Specifies whether to compute the matrix V:\n* = 'V': N columns of V are returned in the array V; Jacobi rotations\n* are not explicitly accumulated.\n* = 'J': N columns of V are returned in the array V, but they are\n* computed as the product of Jacobi rotations. This option is\n* allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.\n* = 'W': V may be used as workspace of length N*N. See the description\n* of V.\n* = 'N': V is not computed.\n* \n* JOBR (input) CHARACTER*1\n* Specifies the RANGE for the singular values. Issues the licence to\n* set to zero small positive singular values if they are outside\n* specified range. If A .NE. 0 is scaled so that the largest singular\n* value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues\n* the licence to kill columns of A whose norm in c*A is less than\n* SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,\n* where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').\n* = 'N': Do not kill small columns of c*A. This option assumes that\n* BLAS and QR factorizations and triangular solvers are\n* implemented to work in that range. If the condition of A\n* is greater than BIG, use SGESVJ.\n* = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]\n* (roughly, as described above). This option is recommended.\n* ===========================\n* For computing the singular values in the FULL range [SFMIN,BIG]\n* use SGESVJ.\n* \n* JOBT (input) CHARACTER*1\n* If the matrix is square then the procedure may determine to use\n* transposed A if A^t seems to be better with respect to convergence.\n* If the matrix is not square, JOBT is ignored. This is subject to\n* changes in the future.\n* The decision is based on two values of entropy over the adjoint\n* orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).\n* = 'T': transpose if entropy test indicates possibly faster\n* convergence of Jacobi process if A^t is taken as input. If A is\n* replaced with A^t, then the row pivoting is included automatically.\n* = 'N': do not speculate.\n* This option can be used to compute only the singular values, or the\n* full SVD (U, SIGMA and V). For only one set of singular vectors\n* (U or V), the caller should provide both U and V, as one of the\n* matrices is used as workspace if the matrix A is transposed.\n* The implementer can easily remove this constraint and make the\n* code more complicated. See the descriptions of U and V.\n* \n* JOBP (input) CHARACTER*1\n* Issues the licence to introduce structured perturbations to drown\n* denormalized numbers. This licence should be active if the\n* denormals are poorly implemented, causing slow computation,\n* especially in cases of fast convergence (!). For details see [1,2].\n* For the sake of simplicity, this perturbations are included only\n* when the full SVD or only the singular values are requested. The\n* implementer/user can easily add the perturbation for the cases of\n* computing one set of singular vectors.\n* = 'P': introduce perturbation\n* = 'N': do not perturb\n*\n* M (input) INTEGER\n* The number of rows of the input matrix A. M >= 0.\n*\n* N (input) INTEGER\n* The number of columns of the input matrix A. M >= N >= 0.\n*\n* A (input/workspace) REAL array, dimension (LDA,N)\n* On entry, the M-by-N matrix A.\n*\n* LDA (input) INTEGER\n* The leading dimension of the array A. LDA >= max(1,M).\n*\n* SVA (workspace/output) REAL array, dimension (N)\n* On exit,\n* - For WORK(1)/WORK(2) = ONE: The singular values of A. During the\n* computation SVA contains Euclidean column norms of the\n* iterated matrices in the array A.\n* - For WORK(1) .NE. WORK(2): The singular values of A are\n* (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if\n* sigma_max(A) overflows or if small singular values have been\n* saved from underflow by scaling the input matrix A.\n* - If JOBR='R' then some of the singular values may be returned\n* as exact zeros obtained by \"set to zero\" because they are\n* below the numerical rank threshold or are denormalized numbers.\n*\n* U (workspace/output) REAL array, dimension ( LDU, N )\n* If JOBU = 'U', then U contains on exit the M-by-N matrix of\n* the left singular vectors.\n* If JOBU = 'F', then U contains on exit the M-by-M matrix of\n* the left singular vectors, including an ONB\n* of the orthogonal complement of the Range(A).\n* If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),\n* then U is used as workspace if the procedure\n* replaces A with A^t. In that case, [V] is computed\n* in U as left singular vectors of A^t and then\n* copied back to the V array. This 'W' option is just\n* a reminder to the caller that in this case U is\n* reserved as workspace of length N*N.\n* If JOBU = 'N' U is not referenced.\n*\n* LDU (input) INTEGER\n* The leading dimension of the array U, LDU >= 1.\n* IF JOBU = 'U' or 'F' or 'W', then LDU >= M.\n*\n* V (workspace/output) REAL array, dimension ( LDV, N )\n* If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of\n* the right singular vectors;\n* If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),\n* then V is used as workspace if the pprocedure\n* replaces A with A^t. In that case, [U] is computed\n* in V as right singular vectors of A^t and then\n* copied back to the U array. This 'W' option is just\n* a reminder to the caller that in this case V is\n* reserved as workspace of length N*N.\n* If JOBV = 'N' V is not referenced.\n*\n* LDV (input) INTEGER\n* The leading dimension of the array V, LDV >= 1.\n* If JOBV = 'V' or 'J' or 'W', then LDV >= N.\n*\n* WORK (workspace/output) REAL array, dimension at least LWORK.\n* On exit,\n* WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such\n* that SCALE*SVA(1:N) are the computed singular values\n* of A. (See the description of SVA().)\n* WORK(2) = See the description of WORK(1).\n* WORK(3) = SCONDA is an estimate for the condition number of\n* column equilibrated A. (If JOBA .EQ. 'E' or 'G')\n* SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).\n* It is computed using SPOCON. It holds\n* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA\n* where R is the triangular factor from the QRF of A.\n* However, if R is truncated and the numerical rank is\n* determined to be strictly smaller than N, SCONDA is\n* returned as -1, thus indicating that the smallest\n* singular values might be lost.\n*\n* If full SVD is needed, the following two condition numbers are\n* useful for the analysis of the algorithm. They are provied for\n* a developer/implementer who is familiar with the details of\n* the method.\n*\n* WORK(4) = an estimate of the scaled condition number of the\n* triangular factor in the first QR factorization.\n* WORK(5) = an estimate of the scaled condition number of the\n* triangular factor in the second QR factorization.\n* The following two parameters are computed if JOBT .EQ. 'T'.\n* They are provided for a developer/implementer who is familiar\n* with the details of the method.\n*\n* WORK(6) = the entropy of A^t*A :: this is the Shannon entropy\n* of diag(A^t*A) / Trace(A^t*A) taken as point in the\n* probability simplex.\n* WORK(7) = the entropy of A*A^t.\n*\n* LWORK (input) INTEGER\n* Length of WORK to confirm proper allocation of work space.\n* LWORK depends on the job:\n*\n* If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and\n* -> .. no scaled condition estimate required ( JOBE.EQ.'N'):\n* LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.\n* For optimal performance (blocked code) the optimal value\n* is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal\n* block size for xGEQP3/xGEQRF.\n* -> .. an estimate of the scaled condition number of A is\n* required (JOBA='E', 'G'). In this case, LWORK is the maximum\n* of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4N,7).\n*\n* If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),\n* -> the minimal requirement is LWORK >= max(2*N+M,7).\n* -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),\n* where NB is the optimal block size.\n*\n* If SIGMA and the left singular vectors are needed\n* -> the minimal requirement is LWORK >= max(2*N+M,7).\n* -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7),\n* where NB is the optimal block size.\n*\n* If full SVD is needed ( JOBU.EQ.'U' or 'F', JOBV.EQ.'V' ) and\n* -> .. the singular vectors are computed without explicit\n* accumulation of the Jacobi rotations, LWORK >= 6*N+2*N*N\n* -> .. in the iterative part, the Jacobi rotations are\n* explicitly accumulated (option, see the description of JOBV),\n* then the minimal requirement is LWORK >= max(M+3*N+N*N,7).\n* For better performance, if NB is the optimal block size,\n* LWORK >= max(3*N+N*N+M,3*N+N*N+N*NB,7).\n*\n* IWORK (workspace/output) INTEGER array, dimension M+3*N.\n* On exit,\n* IWORK(1) = the numerical rank determined after the initial\n* QR factorization with pivoting. See the descriptions\n* of JOBA and JOBR.\n* IWORK(2) = the number of the computed nonzero singular values\n* IWORK(3) = if nonzero, a warning message:\n* If IWORK(3).EQ.1 then some of the column norms of A\n* were denormalized floats. The requested high accuracy\n* is not warranted by the data.\n*\n* INFO (output) INTEGER\n* < 0 : if INFO = -i, then the i-th argument had an illegal value.\n* = 0 : successfull exit;\n* > 0 : SGEJSV did not converge in the maximal allowed number\n* of sweeps. The computed values may be inaccurate.\n*\n\n* Further Details\n* ===============\n*\n* SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,\n* SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an\n* additional row pivoting can be used as a preprocessor, which in some\n* cases results in much higher accuracy. An example is matrix A with the\n* structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned\n* diagonal matrices and C is well-conditioned matrix. In that case, complete\n* pivoting in the first QR factorizations provides accuracy dependent on the\n* condition number of C, and independent of D1, D2. Such higher accuracy is\n* not completely understood theoretically, but it works well in practice.\n* Further, if A can be written as A = B*D, with well-conditioned B and some\n* diagonal D, then the high accuracy is guaranteed, both theoretically and\n* in software, independent of D. For more details see [1], [2].\n* The computational range for the singular values can be the full range\n* ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS\n* & LAPACK routines called by SGEJSV are implemented to work in that range.\n* If that is not the case, then the restriction for safe computation with\n* the singular values in the range of normalized IEEE numbers is that the\n* spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not\n* overflow. This code (SGEJSV) is best used in this restricted range,\n* meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are\n* returned as zeros. See JOBR for details on this.\n* Further, this implementation is somewhat slower than the one described\n* in [1,2] due to replacement of some non-LAPACK components, and because\n* the choice of some tuning parameters in the iterative part (SGESVJ) is\n* left to the implementer on a particular machine.\n* The rank revealing QR factorization (in this code: SGEQP3) should be\n* implemented as in [3]. We have a new version of SGEQP3 under development\n* that is more robust than the current one in LAPACK, with a cleaner cut in\n* rank defficient cases. It will be available in the SIGMA library [4].\n* If M is much larger than N, it is obvious that the inital QRF with\n* column pivoting can be preprocessed by the QRF without pivoting. That\n* well known trick is not used in SGEJSV because in some cases heavy row\n* weighting can be treated with complete pivoting. The overhead in cases\n* M much larger than N is then only due to pivoting, but the benefits in\n* terms of accuracy have prevailed. The implementer/user can incorporate\n* this extra QRF step easily. The implementer can also improve data movement\n* (matrix transpose, matrix copy, matrix transposed copy) - this\n* implementation of SGEJSV uses only the simplest, naive data movement.\n*\n* Contributors\n*\n* Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)\n*\n* References\n*\n* [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.\n* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.\n* LAPACK Working note 169.\n* [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.\n* SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.\n* LAPACK Working note 170.\n* [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR\n* factorization software - a case study.\n* ACM Trans. math. Softw. Vol. 35, No 2 (2008), pp. 1-28.\n* LAPACK Working note 176.\n* [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,\n* QSVD, (H,K)-SVD computations.\n* Department of Mathematics, University of Zagreb, 2008.\n*\n* Bugs, examples and comments\n*\n* Please report all bugs and send interesting examples and/or comments to\n* drmac@math.hr. Thank you.\n*\n* ===========================================================================\n*\n* .. Local Parameters ..\n REAL ZERO, ONE\n PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )\n* ..\n* .. Local Scalars ..\n REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,\n & CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,\n & SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC\n INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING\n LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC,\n & L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,\n & NOSCAL, ROWPIV, RSVEC, TRANSP\n* ..\n* .. Intrinsic Functions ..\n INTRINSIC ABS, ALOG, AMAX1, AMIN1, FLOAT,\n & MAX0, MIN0, NINT, SIGN, SQRT\n* ..\n* .. External Functions ..\n REAL SLAMCH, SNRM2\n INTEGER ISAMAX\n LOGICAL LSAME\n EXTERNAL ISAMAX, LSAME, SLAMCH, SNRM2\n* ..\n* .. External Subroutines ..\n EXTERNAL SCOPY, SGELQF, SGEQP3, SGEQRF, SLACPY, SLASCL,\n & SLASET, SLASSQ, SLASWP, SORGQR, SORMLQ,\n & SORMQR, SPOCON, SSCAL, SSWAP, STRSM, XERBLA\n*\n EXTERNAL SGESVJ\n* ..\n*\n* Test the input arguments\n*\n LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )\n JRACC = LSAME( JOBV, 'J' )\n RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC\n ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )\n L2RANK = LSAME( JOBA, 'R' )\n L2ABER = LSAME( JOBA, 'A' )\n ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )\n L2TRAN = LSAME( JOBT, 'T' )\n L2KILL = LSAME( JOBR, 'R' )\n DEFR = LSAME( JOBR, 'N' )\n L2PERT = LSAME( JOBP, 'P' )\n*\n IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.\n & ERREST .OR. LSAME( JOBA, 'C' ) )) THEN\n INFO = - 1\n ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.\n & LSAME( JOBU, 'W' )) ) THEN\n INFO = - 2\n ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.\n & LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN\n INFO = - 3\n ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN\n INFO = - 4\n ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN\n INFO = - 5\n ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN\n INFO = - 6\n ELSE IF ( M .LT. 0 ) THEN\n INFO = - 7\n ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN\n INFO = - 8\n ELSE IF ( LDA .LT. M ) THEN\n INFO = - 10\n ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN\n INFO = - 13\n ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN\n INFO = - 14\n ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.\n & (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR.\n & (.NOT.(LSVEC .OR. LSVEC) .AND. ERREST .AND.\n & (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR.\n & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.\n & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*N+M))) .OR.\n & (LSVEC .AND. RSVEC .AND. .NOT.JRACC .AND. (LWORK.LT.6*N+2*N*N))\n & .OR. (LSVEC.AND.RSVEC.AND.JRACC.AND.LWORK.LT.MAX0(7,M+3*N+N*N)))\n & THEN\n INFO = - 17\n ELSE\n* #:)\n INFO = 0\n END IF\n*\n IF ( INFO .NE. 0 ) THEN\n* #:(\n CALL XERBLA( 'SGEJSV', - INFO )\n END IF\n*\n* Quick return for void matrix (Y3K safe)\n* #:)\n IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN\n*\n* Determine whether the matrix U should be M x N or M x M\n*\n IF ( LSVEC ) THEN\n N1 = N\n IF ( LSAME( JOBU, 'F' ) ) N1 = M\n END IF\n*\n* Set numerical parameters\n*\n*! NOTE: Make sure SLAMCH() does not fail on the target architecture.\n*\n EPSLN = SLAMCH('Epsilon')\n SFMIN = SLAMCH('SafeMinimum')\n SMALL = SFMIN / EPSLN\n BIG = SLAMCH('O')\n*\n* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N\n*\n*(!) If necessary, scale SVA() to protect the largest norm from\n* overflow. It is possible that this scaling pushes the smallest\n* column norm left from the underflow threshold (extreme case).\n*\n SCALEM = ONE / SQRT(FLOAT(M)*FLOAT(N))\n NOSCAL = .TRUE.\n GOSCAL = .TRUE.\n DO 1874 p = 1, N\n AAPP = ZERO\n AAQQ = ONE\n CALL SLASSQ( M, A(1,p), 1, AAPP, AAQQ )\n IF ( AAPP .GT. BIG ) THEN\n INFO = - 9\n CALL XERBLA( 'SGEJSV', -INFO )\n RETURN\n END IF\n AAQQ = SQRT(AAQQ)\n IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN\n SVA(p) = AAPP * AAQQ\n ELSE\n NOSCAL = .FALSE.\n SVA(p) = AAPP * ( AAQQ * SCALEM )\n IF ( GOSCAL ) THEN\n GOSCAL = .FALSE.\n CALL SSCAL( p-1, SCALEM, SVA, 1 )\n END IF\n END IF\n 1874 CONTINUE\n*\n IF ( NOSCAL ) SCALEM = ONE\n*\n AAPP = ZERO\n AAQQ = BIG\n DO 4781 p = 1, N\n AAPP = AMAX1( AAPP, SVA(p) )\n IF ( SVA(p) .NE. ZERO ) AAQQ = AMIN1( AAQQ, SVA(p) )\n 4781 CONTINUE\n*\n* Quick return for zero M x N matrix\n* #:)\n IF ( AAPP .EQ. ZERO ) THEN\n IF ( LSVEC ) CALL SLASET( 'G', M, N1, ZERO, ONE, U, LDU )\n IF ( RSVEC ) CALL SLASET( 'G', N, N, ZERO, ONE, V, LDV )\n WORK(1) = ONE\n WORK(2) = ONE\n IF ( ERREST ) WORK(3) = ONE\n IF ( LSVEC .AND. RSVEC ) THEN\n WORK(4) = ONE\n WORK(5) = ONE\n END IF\n IF ( L2TRAN ) THEN\n WORK(6) = ZERO\n WORK(7) = ZERO\n END IF\n IWORK(1) = 0\n IWORK(2) = 0\n RETURN\n END IF\n*\n* Issue warning if denormalized column norms detected. Override the\n* high relative accuracy request. Issue licence to kill columns\n* (set them to zero) whose norm is less than sigma_max / BIG (roughly).\n* #:(\n WARNING = 0\n IF ( AAQQ .LE. SFMIN ) THEN\n L2RANK = .TRUE.\n L2KILL = .TRUE.\n WARNING = 1\n END IF\n*\n* Quick return for one-column matrix\n* #:)\n IF ( N .EQ. 1 ) THEN\n*\n IF ( LSVEC ) THEN\n CALL SLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )\n CALL SLACPY( 'A', M, 1, A, LDA, U, LDU )\n* computing all M left singular vectors of the M x 1 matrix\n IF ( N1 .NE. N ) THEN\n CALL SGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR )\n CALL SORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR )\n CALL SCOPY( M, A(1,1), 1, U(1,1), 1 )\n END IF\n END IF\n IF ( RSVEC ) THEN\n V(1,1) = ONE\n END IF\n IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN\n SVA(1) = SVA(1) / SCALEM\n SCALEM = ONE\n END IF\n WORK(1) = ONE / SCALEM\n WORK(2) = ONE\n IF ( SVA(1) .NE. ZERO ) THEN\n IWORK(1) = 1\n IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN\n IWORK(2) = 1\n ELSE\n IWORK(2) = 0\n END IF\n ELSE\n IWORK(1) = 0\n IWORK(2) = 0\n END IF\n IF ( ERREST ) WORK(3) = ONE\n IF ( LSVEC .AND. RSVEC ) THEN\n WORK(4) = ONE\n WORK(5) = ONE\n END IF\n IF ( L2TRAN ) THEN\n WORK(6) = ZERO\n WORK(7) = ZERO\n END IF\n RETURN\n*\n END IF\n*\n TRANSP = .FALSE.\n L2TRAN = L2TRAN .AND. ( M .EQ. N )\n*\n AATMAX = -ONE\n AATMIN = BIG\n IF ( ROWPIV .OR. L2TRAN ) THEN\n*\n* Compute the row norms, needed to determine row pivoting sequence\n* (in the case of heavily row weighted A, row pivoting is strongly\n* advised) and to collect information needed to compare the\n* structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).\n*\n IF ( L2TRAN ) THEN\n DO 1950 p = 1, M\n XSC = ZERO\n TEMP1 = ONE\n CALL SLASSQ( N, A(p,1), LDA, XSC, TEMP1 )\n* SLASSQ gets both the ell_2 and the ell_infinity norm\n* in one pass through the vector\n WORK(M+N+p) = XSC * SCALEM\n WORK(N+p) = XSC * (SCALEM*SQRT(TEMP1))\n AATMAX = AMAX1( AATMAX, WORK(N+p) )\n IF (WORK(N+p) .NE. ZERO) AATMIN = AMIN1(AATMIN,WORK(N+p))\n 1950 CONTINUE\n ELSE\n DO 1904 p = 1, M\n WORK(M+N+p) = SCALEM*ABS( A(p,ISAMAX(N,A(p,1),LDA)) )\n AATMAX = AMAX1( AATMAX, WORK(M+N+p) )\n AATMIN = AMIN1( AATMIN, WORK(M+N+p) )\n 1904 CONTINUE\n END IF\n*\n END IF\n*\n* For square matrix A try to determine whether A^t would be better\n* input for the preconditioned Jacobi SVD, with faster convergence.\n* The decision is based on an O(N) function of the vector of column\n* and row norms of A, based on the Shannon entropy. This should give\n* the right choice in most cases when the difference actually matters.\n* It may fail and pick the slower converging side.\n*\n ENTRA = ZERO\n ENTRAT = ZERO\n IF ( L2TRAN ) THEN\n*\n XSC = ZERO\n TEMP1 = ONE\n CALL SLASSQ( N, SVA, 1, XSC, TEMP1 )\n TEMP1 = ONE / TEMP1\n*\n ENTRA = ZERO\n DO 1113 p = 1, N\n BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1\n IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * ALOG(BIG1)\n 1113 CONTINUE\n ENTRA = - ENTRA / ALOG(FLOAT(N))\n*\n* Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.\n* It is derived from the diagonal of A^t * A. Do the same with the\n* diagonal of A * A^t, compute the entropy of the corresponding\n* probability distribution. Note that A * A^t and A^t * A have the\n* same trace.\n*\n ENTRAT = ZERO\n DO 1114 p = N+1, N+M\n BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1\n IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * ALOG(BIG1)\n 1114 CONTINUE\n ENTRAT = - ENTRAT / ALOG(FLOAT(M))\n*\n* Analyze the entropies and decide A or A^t. Smaller entropy\n* usually means better input for the algorithm.\n*\n TRANSP = ( ENTRAT .LT. ENTRA )\n*\n* If A^t is better than A, transpose A.\n*\n IF ( TRANSP ) THEN\n* In an optimal implementation, this trivial transpose\n* should be replaced with faster transpose.\n DO 1115 p = 1, N - 1\n DO 1116 q = p + 1, N\n TEMP1 = A(q,p)\n A(q,p) = A(p,q)\n A(p,q) = TEMP1\n 1116 CONTINUE\n 1115 CONTINUE\n DO 1117 p = 1, N\n WORK(M+N+p) = SVA(p)\n SVA(p) = WORK(N+p)\n 1117 CONTINUE\n TEMP1 = AAPP\n AAPP = AATMAX\n AATMAX = TEMP1\n TEMP1 = AAQQ\n AAQQ = AATMIN\n AATMIN = TEMP1\n KILL = LSVEC\n LSVEC = RSVEC\n RSVEC = KILL\n IF ( LSVEC ) N1 = N \n*\n ROWPIV = .TRUE.\n END IF\n*\n END IF\n* END IF L2TRAN\n*\n* Scale the matrix so that its maximal singular value remains less\n* than SQRT(BIG) -- the matrix is scaled so that its maximal column\n* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep\n* SQRT(BIG) instead of BIG is the fact that SGEJSV uses LAPACK and\n* BLAS routines that, in some implementations, are not capable of\n* working in the full interval [SFMIN,BIG] and that they may provoke\n* overflows in the intermediate results. If the singular values spread\n* from SFMIN to BIG, then SGESVJ will compute them. So, in that case,\n* one should use SGESVJ instead of SGEJSV.\n*\n BIG1 = SQRT( BIG )\n TEMP1 = SQRT( BIG / FLOAT(N) )\n*\n CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )\n IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN\n AAQQ = ( AAQQ / AAPP ) * TEMP1\n ELSE\n AAQQ = ( AAQQ * TEMP1 ) / AAPP\n END IF\n TEMP1 = TEMP1 * SCALEM\n CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )\n*\n* To undo scaling at the end of this procedure, multiply the\n* computed singular values with USCAL2 / USCAL1.\n*\n USCAL1 = TEMP1\n USCAL2 = AAPP\n*\n IF ( L2KILL ) THEN\n* L2KILL enforces computation of nonzero singular values in\n* the restricted range of condition number of the initial A,\n* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).\n XSC = SQRT( SFMIN )\n ELSE\n XSC = SMALL\n*\n* Now, if the condition number of A is too big,\n* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,\n* as a precaution measure, the full SVD is computed using SGESVJ\n* with accumulated Jacobi rotations. This provides numerically\n* more robust computation, at the cost of slightly increased run\n* time. Depending on the concrete implementation of BLAS and LAPACK\n* (i.e. how they behave in presence of extreme ill-conditioning) the\n* implementor may decide to remove this switch.\n IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN\n JRACC = .TRUE.\n END IF\n*\n END IF\n IF ( AAQQ .LT. XSC ) THEN\n DO 700 p = 1, N\n IF ( SVA(p) .LT. XSC ) THEN\n CALL SLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )\n SVA(p) = ZERO\n END IF\n 700 CONTINUE\n END IF\n*\n* Preconditioning using QR factorization with pivoting\n*\n IF ( ROWPIV ) THEN\n* Optional row permutation (Bjoerck row pivoting):\n* A result by Cox and Higham shows that the Bjoerck's\n* row pivoting combined with standard column pivoting\n* has similar effect as Powell-Reid complete pivoting.\n* The ell-infinity norms of A are made nonincreasing.\n DO 1952 p = 1, M - 1\n q = ISAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1\n IWORK(2*N+p) = q\n IF ( p .NE. q ) THEN\n TEMP1 = WORK(M+N+p)\n WORK(M+N+p) = WORK(M+N+q)\n WORK(M+N+q) = TEMP1\n END IF\n 1952 CONTINUE\n CALL SLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )\n END IF\n*\n* End of the preparation phase (scaling, optional sorting and\n* transposing, optional flushing of small columns).\n*\n* Preconditioning\n*\n* If the full SVD is needed, the right singular vectors are computed\n* from a matrix equation, and for that we need theoretical analysis\n* of the Businger-Golub pivoting. So we use SGEQP3 as the first RR QRF.\n* In all other cases the first RR QRF can be chosen by other criteria\n* (eg speed by replacing global with restricted window pivoting, such\n* as in SGEQPX from TOMS # 782). Good results will be obtained using\n* SGEQPX with properly (!) chosen numerical parameters.\n* Any improvement of SGEQP3 improves overal performance of SGEJSV.\n*\n* A * P1 = Q1 * [ R1^t 0]^t:\n DO 1963 p = 1, N\n* .. all columns are free columns\n IWORK(p) = 0\n 1963 CONTINUE\n CALL SGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )\n*\n* The upper triangular matrix R1 from the first QRF is inspected for\n* rank deficiency and possibilities for deflation, or possible\n* ill-conditioning. Depending on the user specified flag L2RANK,\n* the procedure explores possibilities to reduce the numerical\n* rank by inspecting the computed upper triangular factor. If\n* L2RANK or L2ABER are up, then SGEJSV will compute the SVD of\n* A + dA, where ||dA|| <= f(M,N)*EPSLN.\n*\n NR = 1\n IF ( L2ABER ) THEN\n* Standard absolute error bound suffices. All sigma_i with\n* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an\n* agressive enforcement of lower numerical rank by introducing a\n* backward error of the order of N*EPSLN*||A||.\n TEMP1 = SQRT(FLOAT(N))*EPSLN\n DO 3001 p = 2, N\n IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN\n NR = NR + 1\n ELSE\n GO TO 3002\n END IF\n 3001 CONTINUE\n 3002 CONTINUE\n ELSE IF ( L2RANK ) THEN\n* .. similarly as above, only slightly more gentle (less agressive).\n* Sudden drop on the diagonal of R1 is used as the criterion for\n* close-to-rank-defficient.\n TEMP1 = SQRT(SFMIN)\n DO 3401 p = 2, N\n IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.\n & ( ABS(A(p,p)) .LT. SMALL ) .OR.\n & ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402\n NR = NR + 1\n 3401 CONTINUE\n 3402 CONTINUE\n*\n ELSE\n* The goal is high relative accuracy. However, if the matrix\n* has high scaled condition number the relative accuracy is in\n* general not feasible. Later on, a condition number estimator\n* will be deployed to estimate the scaled condition number.\n* Here we just remove the underflowed part of the triangular\n* factor. This prevents the situation in which the code is\n* working hard to get the accuracy not warranted by the data.\n TEMP1 = SQRT(SFMIN)\n DO 3301 p = 2, N\n IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.\n & ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302\n NR = NR + 1\n 3301 CONTINUE\n 3302 CONTINUE\n*\n END IF\n*\n ALMORT = .FALSE.\n IF ( NR .EQ. N ) THEN\n MAXPRJ = ONE\n DO 3051 p = 2, N\n TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))\n MAXPRJ = AMIN1( MAXPRJ, TEMP1 )\n 3051 CONTINUE\n IF ( MAXPRJ**2 .GE. ONE - FLOAT(N)*EPSLN ) ALMORT = .TRUE.\n END IF\n*\n*\n SCONDA = - ONE\n CONDR1 = - ONE\n CONDR2 = - ONE\n*\n IF ( ERREST ) THEN\n IF ( N .EQ. NR ) THEN\n IF ( RSVEC ) THEN\n* .. V is available as workspace\n CALL SLACPY( 'U', N, N, A, LDA, V, LDV )\n DO 3053 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL SSCAL( p, ONE/TEMP1, V(1,p), 1 )\n 3053 CONTINUE\n CALL SPOCON( 'U', N, V, LDV, ONE, TEMP1,\n & WORK(N+1), IWORK(2*N+M+1), IERR )\n ELSE IF ( LSVEC ) THEN\n* .. U is available as workspace\n CALL SLACPY( 'U', N, N, A, LDA, U, LDU )\n DO 3054 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL SSCAL( p, ONE/TEMP1, U(1,p), 1 )\n 3054 CONTINUE\n CALL SPOCON( 'U', N, U, LDU, ONE, TEMP1,\n & WORK(N+1), IWORK(2*N+M+1), IERR )\n ELSE\n CALL SLACPY( 'U', N, N, A, LDA, WORK(N+1), N )\n DO 3052 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL SSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )\n 3052 CONTINUE\n* .. the columns of R are scaled to have unit Euclidean lengths.\n CALL SPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,\n & WORK(N+N*N+1), IWORK(2*N+M+1), IERR )\n END IF\n SCONDA = ONE / SQRT(TEMP1)\n* SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).\n* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA\n ELSE\n SCONDA = - ONE\n END IF\n END IF\n*\n L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )\n* If there is no violent scaling, artificial perturbation is not needed.\n*\n* Phase 3:\n*\n IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN\n*\n* Singular Values only\n*\n* .. transpose A(1:NR,1:N)\n DO 1946 p = 1, MIN0( N-1, NR )\n CALL SCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )\n 1946 CONTINUE\n*\n* The following two DO-loops introduce small relative perturbation\n* into the strict upper triangle of the lower triangular matrix.\n* Small entries below the main diagonal are also changed.\n* This modification is useful if the computing environment does not\n* provide/allow FLUSH TO ZERO underflow, for it prevents many\n* annoying denormalized numbers in case of strongly scaled matrices.\n* The perturbation is structured so that it does not introduce any\n* new perturbation of the singular values, and it does not destroy\n* the job done by the preconditioner.\n* The licence for this perturbation is in the variable L2PERT, which\n* should be .FALSE. if FLUSH TO ZERO underflow is active.\n*\n IF ( .NOT. ALMORT ) THEN\n*\n IF ( L2PERT ) THEN\n* XSC = SQRT(SMALL)\n XSC = EPSLN / FLOAT(N)\n DO 4947 q = 1, NR\n TEMP1 = XSC*ABS(A(q,q))\n DO 4949 p = 1, N\n IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )\n & .OR. ( p .LT. q ) )\n & A(p,q) = SIGN( TEMP1, A(p,q) )\n 4949 CONTINUE\n 4947 CONTINUE\n ELSE\n CALL SLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )\n END IF\n*\n* .. second preconditioning using the QR factorization\n*\n CALL SGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )\n*\n* .. and transpose upper to lower triangular\n DO 1948 p = 1, NR - 1\n CALL SCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )\n 1948 CONTINUE\n*\n END IF\n*\n* Row-cyclic Jacobi SVD algorithm with column pivoting\n*\n* .. again some perturbation (a \"background noise\") is added\n* to drown denormals\n IF ( L2PERT ) THEN\n* XSC = SQRT(SMALL)\n XSC = EPSLN / FLOAT(N)\n DO 1947 q = 1, NR\n TEMP1 = XSC*ABS(A(q,q))\n DO 1949 p = 1, NR\n IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )\n & .OR. ( p .LT. q ) )\n & A(p,q) = SIGN( TEMP1, A(p,q) )\n 1949 CONTINUE\n 1947 CONTINUE\n ELSE\n CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2), LDA )\n END IF\n*\n* .. and one-sided Jacobi rotations are started on a lower\n* triangular matrix (plus perturbation which is ignored in\n* the part which destroys triangular form (confusing?!))\n*\n CALL SGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,\n & N, V, LDV, WORK, LWORK, INFO )\n*\n SCALEM = WORK(1)\n NUMRANK = NINT(WORK(2))\n*\n*\n ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN\n*\n* -> Singular Values and Right Singular Vectors <-\n*\n IF ( ALMORT ) THEN\n*\n* .. in this case NR equals N\n DO 1998 p = 1, NR\n CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 1998 CONTINUE\n CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n*\n CALL SGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,\n & WORK, LWORK, INFO )\n SCALEM = WORK(1)\n NUMRANK = NINT(WORK(2))\n\n ELSE\n*\n* .. two more QR factorizations ( one QRF is not enough, two require\n* accumulated product of Jacobi rotations, three are perfect )\n*\n CALL SLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA )\n CALL SGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR)\n CALL SLACPY( 'Lower', NR, NR, A, LDA, V, LDV )\n CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n CALL SGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n DO 8998 p = 1, NR\n CALL SCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )\n 8998 CONTINUE\n CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n*\n CALL SGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,\n & LDU, WORK(N+1), LWORK, INFO )\n SCALEM = WORK(N+1)\n NUMRANK = NINT(WORK(N+2))\n IF ( NR .LT. N ) THEN\n CALL SLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1), LDV )\n CALL SLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1), LDV )\n CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV )\n END IF\n*\n CALL SORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,\n & V, LDV, WORK(N+1), LWORK-N, IERR )\n*\n END IF\n*\n DO 8991 p = 1, N\n CALL SCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )\n 8991 CONTINUE\n CALL SLACPY( 'All', N, N, A, LDA, V, LDV )\n*\n IF ( TRANSP ) THEN\n CALL SLACPY( 'All', N, N, V, LDV, U, LDU )\n END IF\n*\n ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN\n*\n* .. Singular Values and Left Singular Vectors ..\n*\n* .. second preconditioning step to avoid need to accumulate\n* Jacobi rotations in the Jacobi iterations.\n DO 1965 p = 1, NR\n CALL SCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )\n 1965 CONTINUE\n CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n*\n CALL SGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n*\n DO 1967 p = 1, NR - 1\n CALL SCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )\n 1967 CONTINUE\n CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n*\n CALL SGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,\n & LDA, WORK(N+1), LWORK-N, INFO )\n SCALEM = WORK(N+1)\n NUMRANK = NINT(WORK(N+2))\n*\n IF ( NR .LT. M ) THEN\n CALL SLASET( 'A', M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL SLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU )\n CALL SLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU )\n END IF\n END IF\n*\n CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n*\n IF ( ROWPIV )\n & CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n*\n DO 1974 p = 1, N1\n XSC = ONE / SNRM2( M, U(1,p), 1 )\n CALL SSCAL( M, XSC, U(1,p), 1 )\n 1974 CONTINUE\n*\n IF ( TRANSP ) THEN\n CALL SLACPY( 'All', N, N, U, LDU, V, LDV )\n END IF\n*\n ELSE\n*\n* .. Full SVD ..\n*\n IF ( .NOT. JRACC ) THEN\n*\n IF ( .NOT. ALMORT ) THEN\n*\n* Second Preconditioning Step (QRF [with pivoting])\n* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is\n* equivalent to an LQF CALL. Since in many libraries the QRF\n* seems to be better optimized than the LQF, we do explicit\n* transpose and use the QRF. This is subject to changes in an\n* optimized implementation of SGEJSV.\n*\n DO 1968 p = 1, NR\n CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 1968 CONTINUE\n*\n* .. the following two loops perturb small entries to avoid\n* denormals in the second QR factorization, where they are\n* as good as zeros. This is done to avoid painfully slow\n* computation with denormals. The relative size of the perturbation\n* is a parameter that can be changed by the implementer.\n* This perturbation device will be obsolete on machines with\n* properly implemented arithmetic.\n* To switch it off, set L2PERT=.FALSE. To remove it from the\n* code, remove the action under L2PERT=.TRUE., leave the ELSE part.\n* The following two loops should be blocked and fused with the\n* transposed copy above.\n*\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)\n DO 2969 q = 1, NR\n TEMP1 = XSC*ABS( V(q,q) )\n DO 2968 p = 1, N\n IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )\n & .OR. ( p .LT. q ) )\n & V(p,q) = SIGN( TEMP1, V(p,q) )\n IF ( p. LT. q ) V(p,q) = - V(p,q)\n 2968 CONTINUE\n 2969 CONTINUE\n ELSE\n CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n END IF\n*\n* Estimate the row scaled condition number of R1\n* (If R1 is rectangular, N > NR, then the condition number\n* of the leading NR x NR submatrix is estimated.)\n*\n CALL SLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )\n DO 3950 p = 1, NR\n TEMP1 = SNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)\n CALL SSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)\n 3950 CONTINUE\n CALL SPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,\n & WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)\n CONDR1 = ONE / SQRT(TEMP1)\n* .. here need a second oppinion on the condition number\n* .. then assume worst case scenario\n* R1 is OK for inverse <=> CONDR1 .LT. FLOAT(N)\n* more conservative <=> CONDR1 .LT. SQRT(FLOAT(N))\n*\n COND_OK = SQRT(FLOAT(NR))\n*[TP] COND_OK is a tuning parameter.\n\n IF ( CONDR1 .LT. COND_OK ) THEN\n* .. the second QRF without pivoting. Note: in an optimized\n* implementation, this QRF should be implemented as the QRF\n* of a lower triangular matrix.\n* R1^t = Q2 * R2\n CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n*\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)/EPSLN\n DO 3959 p = 2, NR\n DO 3958 q = 1, p - 1\n TEMP1 = XSC * AMIN1(ABS(V(p,p)),ABS(V(q,q)))\n IF ( ABS(V(q,p)) .LE. TEMP1 )\n & V(q,p) = SIGN( TEMP1, V(q,p) )\n 3958 CONTINUE\n 3959 CONTINUE\n END IF\n*\n IF ( NR .NE. N )\n* .. save ...\n & CALL SLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )\n*\n* .. this transposed copy should be better than naive\n DO 1969 p = 1, NR - 1\n CALL SCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )\n 1969 CONTINUE\n*\n CONDR2 = CONDR1\n*\n ELSE\n*\n* .. ill-conditioned case: second QRF with pivoting\n* Note that windowed pivoting would be equaly good\n* numerically, and more run-time efficient. So, in\n* an optimal implementation, the next call to SGEQP3\n* should be replaced with eg. CALL SGEQPX (ACM TOMS #782)\n* with properly (carefully) chosen parameters.\n*\n* R1^t * P2 = Q2 * R2\n DO 3003 p = 1, NR\n IWORK(N+p) = 0\n 3003 CONTINUE\n CALL SGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),\n & WORK(2*N+1), LWORK-2*N, IERR )\n** CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n** & LWORK-2*N, IERR )\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)\n DO 3969 p = 2, NR\n DO 3968 q = 1, p - 1\n TEMP1 = XSC * AMIN1(ABS(V(p,p)),ABS(V(q,q)))\n IF ( ABS(V(q,p)) .LE. TEMP1 )\n & V(q,p) = SIGN( TEMP1, V(q,p) )\n 3968 CONTINUE\n 3969 CONTINUE\n END IF\n*\n CALL SLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )\n*\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)\n DO 8970 p = 2, NR\n DO 8971 q = 1, p - 1\n TEMP1 = XSC * AMIN1(ABS(V(p,p)),ABS(V(q,q)))\n V(p,q) = - SIGN( TEMP1, V(q,p) )\n 8971 CONTINUE\n 8970 CONTINUE\n ELSE\n CALL SLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )\n END IF\n* Now, compute R2 = L3 * Q3, the LQ factorization.\n CALL SGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),\n & WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )\n* .. and estimate the condition number\n CALL SLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )\n DO 4950 p = 1, NR\n TEMP1 = SNRM2( p, WORK(2*N+N*NR+NR+p), NR )\n CALL SSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )\n 4950 CONTINUE\n CALL SPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,\n & WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )\n CONDR2 = ONE / SQRT(TEMP1)\n*\n IF ( CONDR2 .GE. COND_OK ) THEN\n* .. save the Householder vectors used for Q3\n* (this overwrittes the copy of R2, as it will not be\n* needed in this branch, but it does not overwritte the\n* Huseholder vectors of Q2.).\n CALL SLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )\n* .. and the rest of the information on Q3 is in\n* WORK(2*N+N*NR+1:2*N+N*NR+N)\n END IF\n*\n END IF\n*\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)\n DO 4968 q = 2, NR\n TEMP1 = XSC * V(q,q)\n DO 4969 p = 1, q - 1\n* V(p,q) = - SIGN( TEMP1, V(q,p) )\n V(p,q) = - SIGN( TEMP1, V(p,q) )\n 4969 CONTINUE\n 4968 CONTINUE\n ELSE\n CALL SLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )\n END IF\n*\n* Second preconditioning finished; continue with Jacobi SVD\n* The input matrix is lower trinagular.\n*\n* Recover the right singular vectors as solution of a well\n* conditioned triangular matrix equation.\n*\n IF ( CONDR1 .LT. COND_OK ) THEN\n*\n CALL SGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,\n & LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )\n SCALEM = WORK(2*N+N*NR+NR+1)\n NUMRANK = NINT(WORK(2*N+N*NR+NR+2))\n DO 3970 p = 1, NR\n CALL SCOPY( NR, V(1,p), 1, U(1,p), 1 )\n CALL SSCAL( NR, SVA(p), V(1,p), 1 )\n 3970 CONTINUE\n\n* .. pick the right matrix equation and solve it\n*\n IF ( NR. EQ. N ) THEN\n* :)) .. best case, R1 is inverted. The solution of this matrix\n* equation is Q2*V2 = the product of the Jacobi rotations\n* used in SGESVJ, premultiplied with the orthogonal matrix\n* from the second QR factorization.\n CALL STRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,LDV )\n ELSE\n* .. R1 is well conditioned, but non-square. Transpose(R2)\n* is inverted to get the product of the Jacobi rotations\n* used in SGESVJ. The Q-factor from the second QR\n* factorization is then built in explicitly.\n CALL STRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),\n & N,V,LDV)\n IF ( NR .LT. N ) THEN\n CALL SLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)\n CALL SLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)\n CALL SLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)\n END IF\n CALL SORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n & V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)\n END IF\n*\n ELSE IF ( CONDR2 .LT. COND_OK ) THEN\n*\n* :) .. the input matrix A is very likely a relative of\n* the Kahan matrix :)\n* The matrix R2 is inverted. The solution of the matrix equation\n* is Q3^T*V3 = the product of the Jacobi rotations (appplied to\n* the lower triangular L3 from the LQ factorization of\n* R2=L3*Q3), pre-multiplied with the transposed Q3.\n CALL SGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,\n & LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )\n SCALEM = WORK(2*N+N*NR+NR+1)\n NUMRANK = NINT(WORK(2*N+N*NR+NR+2))\n DO 3870 p = 1, NR\n CALL SCOPY( NR, V(1,p), 1, U(1,p), 1 )\n CALL SSCAL( NR, SVA(p), U(1,p), 1 )\n 3870 CONTINUE\n CALL STRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,LDU)\n* .. apply the permutation from the second QR factorization\n DO 873 q = 1, NR\n DO 872 p = 1, NR\n WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)\n 872 CONTINUE\n DO 874 p = 1, NR\n U(p,q) = WORK(2*N+N*NR+NR+p)\n 874 CONTINUE\n 873 CONTINUE\n IF ( NR .LT. N ) THEN\n CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n & V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n ELSE\n* Last line of defense.\n* #:( This is a rather pathological case: no scaled condition\n* improvement after two pivoted QR factorizations. Other\n* possibility is that the rank revealing QR factorization\n* or the condition estimator has failed, or the COND_OK\n* is set very close to ONE (which is unnecessary). Normally,\n* this branch should never be executed, but in rare cases of\n* failure of the RRQR or condition estimator, the last line of\n* defense ensures that SGEJSV completes the task.\n* Compute the full SVD of L3 using SGESVJ with explicit\n* accumulation of Jacobi rotations.\n CALL SGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,\n & LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )\n SCALEM = WORK(2*N+N*NR+NR+1)\n NUMRANK = NINT(WORK(2*N+N*NR+NR+2))\n IF ( NR .LT. N ) THEN\n CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n & V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n*\n CALL SORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,\n & WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),\n & LWORK-2*N-N*NR-NR, IERR )\n DO 773 q = 1, NR\n DO 772 p = 1, NR\n WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)\n 772 CONTINUE\n DO 774 p = 1, NR\n U(p,q) = WORK(2*N+N*NR+NR+p)\n 774 CONTINUE\n 773 CONTINUE\n*\n END IF\n*\n* Permute the rows of V using the (column) permutation from the\n* first QRF. Also, scale the columns to make them unit in\n* Euclidean norm. This applies to all cases.\n*\n TEMP1 = SQRT(FLOAT(N)) * EPSLN\n DO 1972 q = 1, N\n DO 972 p = 1, N\n WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)\n 972 CONTINUE\n DO 973 p = 1, N\n V(p,q) = WORK(2*N+N*NR+NR+p)\n 973 CONTINUE\n XSC = ONE / SNRM2( N, V(1,q), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL SSCAL( N, XSC, V(1,q), 1 )\n 1972 CONTINUE\n* At this moment, V contains the right singular vectors of A.\n* Next, assemble the left singular vector matrix U (M x N).\n IF ( NR .LT. M ) THEN\n CALL SLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL SLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)\n CALL SLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),LDU)\n END IF\n END IF\n*\n* The Q matrix from the first QRF is built into the left singular\n* matrix U. This applies to all cases.\n*\n CALL SORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n\n* The columns of U are normalized. The cost is O(M*N) flops.\n TEMP1 = SQRT(FLOAT(M)) * EPSLN\n DO 1973 p = 1, NR\n XSC = ONE / SNRM2( M, U(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL SSCAL( M, XSC, U(1,p), 1 )\n 1973 CONTINUE\n*\n* If the initial QRF is computed with row pivoting, the left\n* singular vectors must be adjusted.\n*\n IF ( ROWPIV )\n & CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n*\n ELSE\n*\n* .. the initial matrix A has almost orthogonal columns and\n* the second QRF is not needed\n*\n CALL SLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL)\n DO 5970 p = 2, N\n TEMP1 = XSC * WORK( N + (p-1)*N + p )\n DO 5971 q = 1, p - 1\n WORK(N+(q-1)*N+p)=-SIGN(TEMP1,WORK(N+(p-1)*N+q))\n 5971 CONTINUE\n 5970 CONTINUE\n ELSE\n CALL SLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )\n END IF\n*\n CALL SGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,\n & N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )\n*\n SCALEM = WORK(N+N*N+1)\n NUMRANK = NINT(WORK(N+N*N+2))\n DO 6970 p = 1, N\n CALL SCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )\n CALL SSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )\n 6970 CONTINUE\n*\n CALL STRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,\n & ONE, A, LDA, WORK(N+1), N )\n DO 6972 p = 1, N\n CALL SCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )\n 6972 CONTINUE\n TEMP1 = SQRT(FLOAT(N))*EPSLN\n DO 6971 p = 1, N\n XSC = ONE / SNRM2( N, V(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL SSCAL( N, XSC, V(1,p), 1 )\n 6971 CONTINUE\n*\n* Assemble the left singular vector matrix U (M x N).\n*\n IF ( N .LT. M ) THEN\n CALL SLASET( 'A', M-N, N, ZERO, ZERO, U(N+1,1), LDU )\n IF ( N .LT. N1 ) THEN\n CALL SLASET( 'A',N, N1-N, ZERO, ZERO, U(1,N+1),LDU )\n CALL SLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU )\n END IF\n END IF\n CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n TEMP1 = SQRT(FLOAT(M))*EPSLN\n DO 6973 p = 1, N1\n XSC = ONE / SNRM2( M, U(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL SSCAL( M, XSC, U(1,p), 1 )\n 6973 CONTINUE\n*\n IF ( ROWPIV )\n & CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n*\n END IF\n*\n* end of the >> almost orthogonal case << in the full SVD\n*\n ELSE\n*\n* This branch deploys a preconditioned Jacobi SVD with explicitly\n* accumulated rotations. It is included as optional, mainly for\n* experimental purposes. It does perfom well, and can also be used.\n* In this implementation, this branch will be automatically activated\n* if the condition number sigma_max(A) / sigma_min(A) is predicted\n* to be greater than the overflow threshold. This is because the\n* a posteriori computation of the singular vectors assumes robust\n* implementation of BLAS and some LAPACK procedures, capable of working\n* in presence of extreme values. Since that is not always the case, ...\n*\n DO 7968 p = 1, NR\n CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 7968 CONTINUE\n*\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL/EPSLN)\n DO 5969 q = 1, NR\n TEMP1 = XSC*ABS( V(q,q) )\n DO 5968 p = 1, N\n IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )\n & .OR. ( p .LT. q ) )\n & V(p,q) = SIGN( TEMP1, V(p,q) )\n IF ( p. LT. q ) V(p,q) = - V(p,q)\n 5968 CONTINUE\n 5969 CONTINUE\n ELSE\n CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n END IF\n\n CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n CALL SLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )\n*\n DO 7969 p = 1, NR\n CALL SCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )\n 7969 CONTINUE\n\n IF ( L2PERT ) THEN\n XSC = SQRT(SMALL/EPSLN)\n DO 9970 q = 2, NR\n DO 9971 p = 1, q - 1\n TEMP1 = XSC * AMIN1(ABS(U(p,p)),ABS(U(q,q)))\n U(p,q) = - SIGN( TEMP1, U(q,p) )\n 9971 CONTINUE\n 9970 CONTINUE\n ELSE\n CALL SLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n END IF\n\n CALL SGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,\n & N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )\n SCALEM = WORK(2*N+N*NR+1)\n NUMRANK = NINT(WORK(2*N+N*NR+2))\n\n IF ( NR .LT. N ) THEN\n CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n\n CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),\n & V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )\n*\n* Permute the rows of V using the (column) permutation from the\n* first QRF. Also, scale the columns to make them unit in\n* Euclidean norm. This applies to all cases.\n*\n TEMP1 = SQRT(FLOAT(N)) * EPSLN\n DO 7972 q = 1, N\n DO 8972 p = 1, N\n WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)\n 8972 CONTINUE\n DO 8973 p = 1, N\n V(p,q) = WORK(2*N+N*NR+NR+p)\n 8973 CONTINUE\n XSC = ONE / SNRM2( N, V(1,q), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL SSCAL( N, XSC, V(1,q), 1 )\n 7972 CONTINUE\n*\n* At this moment, V contains the right singular vectors of A.\n* Next, assemble the left singular vector matrix U (M x N).\n*\n IF ( NR .LT. M ) THEN\n CALL SLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL SLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),LDU )\n CALL SLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU )\n END IF\n END IF\n*\n CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n*\n IF ( ROWPIV )\n & CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n*\n*\n END IF\n IF ( TRANSP ) THEN\n* .. swap U and V because the procedure worked on A^t\n DO 6974 p = 1, N\n CALL SSWAP( N, U(1,p), 1, V(1,p), 1 )\n 6974 CONTINUE\n END IF\n*\n END IF\n* end of the full SVD\n*\n* Undo scaling, if necessary (and possible)\n*\n IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN\n CALL SLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )\n USCAL1 = ONE\n USCAL2 = ONE\n END IF\n*\n IF ( NR .LT. N ) THEN\n DO 3004 p = NR+1, N\n SVA(p) = ZERO\n 3004 CONTINUE\n END IF\n*\n WORK(1) = USCAL2 * SCALEM\n WORK(2) = USCAL1\n IF ( ERREST ) WORK(3) = SCONDA\n IF ( LSVEC .AND. RSVEC ) THEN\n WORK(4) = CONDR1\n WORK(5) = CONDR2\n END IF\n IF ( L2TRAN ) THEN\n WORK(6) = ENTRA\n WORK(7) = ENTRAT\n END IF\n*\n IWORK(1) = NR\n IWORK(2) = NUMRANK\n IWORK(3) = WARNING\n*\n RETURN\n* ..\n* .. END OF SGEJSV\n* ..\n END\n*\n\n");
return Qnil;
}
if (rb_hash_aref(rblapack_options, sUsage) == Qtrue) {
printf("%s\n", "USAGE:\n sva, u, v, iwork, info, work = NumRu::Lapack.sgejsv( joba, jobu, jobv, jobr, jobt, jobp, m, a, work, [:lwork => lwork, :usage => usage, :help => help])\n");
return Qnil;
}
} else
rblapack_options = Qnil;
if (argc != 9 && argc != 10)
rb_raise(rb_eArgError,"wrong number of arguments (%d for 9)", argc);
rblapack_joba = argv[0];
rblapack_jobu = argv[1];
rblapack_jobv = argv[2];
rblapack_jobr = argv[3];
rblapack_jobt = argv[4];
rblapack_jobp = argv[5];
rblapack_m = argv[6];
rblapack_a = argv[7];
rblapack_work = argv[8];
if (argc == 10) {
rblapack_lwork = argv[9];
} else if (rblapack_options != Qnil) {
rblapack_lwork = rb_hash_aref(rblapack_options, ID2SYM(rb_intern("lwork")));
} else {
rblapack_lwork = Qnil;
}
joba = StringValueCStr(rblapack_joba)[0];
jobv = StringValueCStr(rblapack_jobv)[0];
jobt = StringValueCStr(rblapack_jobt)[0];
m = NUM2INT(rblapack_m);
if (!NA_IsNArray(rblapack_work))
rb_raise(rb_eArgError, "work (9th argument) must be NArray");
if (NA_RANK(rblapack_work) != 1)
rb_raise(rb_eArgError, "rank of work (9th argument) must be %d", 1);
lwork = NA_SHAPE0(rblapack_work);
if (NA_TYPE(rblapack_work) != NA_SFLOAT)
rblapack_work = na_change_type(rblapack_work, NA_SFLOAT);
work = NA_PTR_TYPE(rblapack_work, real*);
jobu = StringValueCStr(rblapack_jobu)[0];
jobp = StringValueCStr(rblapack_jobp)[0];
ldu = (lsame_(&jobu,"U")||lsame_(&jobu,"F")||lsame_(&jobu,"W")) ? m : 1;
jobr = StringValueCStr(rblapack_jobr)[0];
if (!NA_IsNArray(rblapack_a))
rb_raise(rb_eArgError, "a (8th argument) must be NArray");
if (NA_RANK(rblapack_a) != 2)
rb_raise(rb_eArgError, "rank of a (8th argument) must be %d", 2);
lda = NA_SHAPE0(rblapack_a);
n = NA_SHAPE1(rblapack_a);
if (NA_TYPE(rblapack_a) != NA_SFLOAT)
rblapack_a = na_change_type(rblapack_a, NA_SFLOAT);
a = NA_PTR_TYPE(rblapack_a, real*);
ldv = (lsame_(&jobu,"U")||lsame_(&jobu,"F")||lsame_(&jobu,"W")) ? n : 1;
if (rblapack_lwork == Qnil)
lwork = (lsame_(&jobu,"N")&&lsame_(&jobv,"N")) ? MAX(MAX(2*m+n,4*n+n*n),7) : lsame_(&jobv,"V") ? MAX(2*n+m,7) : ((lsame_(&jobu,"U")||lsame_(&jobu,"F"))&&lsame_(&jobv,"V")) ? MAX(MAX(6*n+2*n*n,m+3*n+n*n),7) : MAX(2*n+m,7);
else {
lwork = NUM2INT(rblapack_lwork);
}
{
na_shape_t shape[1];
shape[0] = n;
rblapack_sva = na_make_object(NA_SFLOAT, 1, shape, cNArray);
}
sva = NA_PTR_TYPE(rblapack_sva, real*);
{
na_shape_t shape[2];
shape[0] = ldu;
shape[1] = n;
rblapack_u = na_make_object(NA_SFLOAT, 2, shape, cNArray);
}
u = NA_PTR_TYPE(rblapack_u, real*);
{
na_shape_t shape[2];
shape[0] = ldv;
shape[1] = n;
rblapack_v = na_make_object(NA_SFLOAT, 2, shape, cNArray);
}
v = NA_PTR_TYPE(rblapack_v, real*);
{
na_shape_t shape[1];
shape[0] = m+3*n;
rblapack_iwork = na_make_object(NA_LINT, 1, shape, cNArray);
}
iwork = NA_PTR_TYPE(rblapack_iwork, integer*);
{
na_shape_t shape[1];
shape[0] = lwork;
rblapack_work_out__ = na_make_object(NA_SFLOAT, 1, shape, cNArray);
}
work_out__ = NA_PTR_TYPE(rblapack_work_out__, real*);
MEMCPY(work_out__, work, real, NA_TOTAL(rblapack_work));
rblapack_work = rblapack_work_out__;
work = work_out__;
sgejsv_(&joba, &jobu, &jobv, &jobr, &jobt, &jobp, &m, &n, a, &lda, sva, u, &ldu, v, &ldv, work, &lwork, iwork, &info);
rblapack_info = INT2NUM(info);
return rb_ary_new3(6, rblapack_sva, rblapack_u, rblapack_v, rblapack_iwork, rblapack_info, rblapack_work);
}
void
init_lapack_sgejsv(VALUE mLapack, VALUE sH, VALUE sU, VALUE zero){
sHelp = sH;
sUsage = sU;
rblapack_ZERO = zero;
rb_define_module_function(mLapack, "sgejsv", rblapack_sgejsv, -1);
}
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