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---
:name: dgejsv
:md5sum: 48e427f846d8774d81a35aecade0f887
:category: :subroutine
:arguments:
- joba:
:type: char
:intent: input
- jobu:
:type: char
:intent: input
- jobv:
:type: char
:intent: input
- jobr:
:type: char
:intent: input
- jobt:
:type: char
:intent: input
- jobp:
:type: char
:intent: input
- m:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- a:
:type: doublereal
:intent: input
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- sva:
:type: doublereal
:intent: output
:dims:
- n
- u:
:type: doublereal
:intent: output
:dims:
- ldu
- n
- ldu:
:type: integer
:intent: input
- v:
:type: doublereal
:intent: output
:dims:
- ldv
- n
- ldv:
:type: integer
:intent: input
- work:
:type: doublereal
:intent: input/output
:dims:
- lwork
- lwork:
:type: integer
:intent: input
:option: true
:default: "(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)"
- iwork:
:type: integer
:intent: output
:dims:
- m+3*n
- info:
:type: integer
:intent: output
:substitutions:
ldu: "(lsame_(&jobu,\"U\")||lsame_(&jobu,\"F\")||lsame_(&jobu,\"W\")) ? m : 1"
ldv: "(lsame_(&jobu,\"U\")||lsame_(&jobu,\"F\")||lsame_(&jobu,\"W\")) ? n : 1"
:fortran_help: " SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DGEJSV 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 DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues\n\
* the licence to kill columns of A whose norm in c*A is less than\n\
* DSQRT(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 DGESVJ.\n\
* = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(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 DGESVJ.\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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 DSQRT(||(R^t * R)^(-1)||_1).\n\
* It is computed using DPOCON. 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 : DGEJSV 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\
* DGEJSV 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 DGEJSV 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 (DGEJSV) 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 (DGESVJ) 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 DGEJSV 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 DGEJSV 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 DOUBLE PRECISION ZERO, ONE\n PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )\n\
* ..\n\
* .. Local Scalars ..\n DOUBLE PRECISION 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 DABS, DLOG, DMAX1, DMIN1, DBLE,\n & MAX0, MIN0, IDNINT, DSIGN, DSQRT\n\
* ..\n\
* .. External Functions ..\n DOUBLE PRECISION DLAMCH, DNRM2\n INTEGER IDAMAX\n LOGICAL LSAME\n EXTERNAL IDAMAX, LSAME, DLAMCH, DNRM2\n\
* ..\n\
* .. External Subroutines ..\n EXTERNAL DCOPY, DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL,\n & DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,\n & DORMQR, DPOCON, DSCAL, DSWAP, DTRSM, XERBLA\n\
*\n EXTERNAL DGESVJ\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( 'DGEJSV', - 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 DLAMCH() does not fail on the target architecture.\n\
*\n\n EPSLN = DLAMCH('Epsilon')\n SFMIN = DLAMCH('SafeMinimum')\n SMALL = SFMIN / EPSLN\n BIG = DLAMCH('O')\n\
* BIG = ONE / SFMIN\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 / DSQRT(DBLE(M)*DBLE(N))\n NOSCAL = .TRUE.\n GOSCAL = .TRUE.\n DO 1874 p = 1, N\n AAPP = ZERO\n AAQQ = ONE\n CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ )\n IF ( AAPP .GT. BIG ) THEN\n INFO = - 9\n CALL XERBLA( 'DGEJSV', -INFO )\n RETURN\n END IF\n AAQQ = DSQRT(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 DSCAL( 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 = DMAX1( AAPP, SVA(p) )\n IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( 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 DLASET( 'G', M, N1, ZERO, ONE, U, LDU )\n IF ( RSVEC ) CALL DLASET( '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 DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )\n CALL DLACPY( '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 DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR )\n CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR )\n CALL DCOPY( 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 DLASSQ( N, A(p,1), LDA, XSC, TEMP1 )\n\
* DLASSQ 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*DSQRT(TEMP1))\n AATMAX = DMAX1( AATMAX, WORK(N+p) )\n IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p))\n 1950 CONTINUE\n ELSE\n DO 1904 p = 1, M\n WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )\n AATMAX = DMAX1( AATMAX, WORK(M+N+p) )\n AATMIN = DMIN1( 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 DLASSQ( 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 * DLOG(BIG1)\n 1113 CONTINUE\n ENTRA = - ENTRA / DLOG(DBLE(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 * DLOG(BIG1)\n 1114 CONTINUE\n ENTRAT = - ENTRAT / DLOG(DBLE(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 DSQRT(BIG) -- the matrix is scaled so that its maximal column\n\
* has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep\n\
* DSQRT(BIG) instead of BIG is the fact that DGEJSV 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 DGESVJ will compute them. So, in that case,\n\
* one should use DGESVJ instead of DGEJSV.\n\
*\n BIG1 = DSQRT( BIG )\n TEMP1 = DSQRT( BIG / DBLE(N) )\n\
*\n CALL DLASCL( '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 DLASCL( '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. DSQRT(BIG)/DSQRT(SFMIN).\n XSC = DSQRT( 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. DSQRT(BIG/N) * EPSLN / SFMIN,\n\
* as a precaution measure, the full SVD is computed using DGESVJ\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.DSQRT(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 DLASET( '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 = IDAMAX( 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 DLASWP( 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 DGEQP3 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 DGEQP3 improves overal performance of DGEJSV.\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 DGEQP3( 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 DGEJSV 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 = DSQRT(DBLE(N))*EPSLN\n DO 3001 p = 2, N\n IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(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 = DSQRT(SFMIN)\n DO 3401 p = 2, N\n IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.\n & ( DABS(A(p,p)) .LT. SMALL ) .OR.\n & ( L2KILL .AND. (DABS(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 = DSQRT(SFMIN)\n DO 3301 p = 2, N\n IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.\n & ( L2KILL .AND. (DABS(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 = DABS(A(p,p)) / SVA(IWORK(p))\n MAXPRJ = DMIN1( MAXPRJ, TEMP1 )\n 3051 CONTINUE\n IF ( MAXPRJ**2 .GE. ONE - DBLE(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 DLACPY( 'U', N, N, A, LDA, V, LDV )\n DO 3053 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )\n 3053 CONTINUE\n CALL DPOCON( '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 DLACPY( 'U', N, N, A, LDA, U, LDU )\n DO 3054 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )\n 3054 CONTINUE\n CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,\n & WORK(N+1), IWORK(2*N+M+1), IERR )\n ELSE\n CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )\n DO 3052 p = 1, N\n TEMP1 = SVA(IWORK(p))\n CALL DSCAL( 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 DPOCON( '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 / DSQRT(TEMP1)\n\
* SCONDA is an estimate of DSQRT(||(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. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )\n\
* If there is no violent scaling, artificial perturbation is not needed.\n\
*\n\
* Phase 3:\n\
*\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 DCOPY( 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 = DSQRT(SMALL)\n XSC = EPSLN / DBLE(N)\n DO 4947 q = 1, NR\n TEMP1 = XSC*DABS(A(q,q))\n DO 4949 p = 1, N\n IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )\n & .OR. ( p .LT. q ) )\n & A(p,q) = DSIGN( TEMP1, A(p,q) )\n 4949 CONTINUE\n 4947 CONTINUE\n ELSE\n CALL DLASET( '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 DGEQRF( 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 DCOPY( 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 = DSQRT(SMALL)\n XSC = EPSLN / DBLE(N)\n DO 1947 q = 1, NR\n TEMP1 = XSC*DABS(A(q,q))\n DO 1949 p = 1, NR\n IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )\n & .OR. ( p .LT. q ) )\n & A(p,q) = DSIGN( TEMP1, A(p,q) )\n 1949 CONTINUE\n 1947 CONTINUE\n ELSE\n CALL DLASET( '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 DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,\n & N, V, LDV, WORK, LWORK, INFO )\n\
*\n SCALEM = WORK(1)\n NUMRANK = IDNINT(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 DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 1998 CONTINUE\n CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n\
*\n CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,\n & WORK, LWORK, INFO )\n SCALEM = WORK(1)\n NUMRANK = IDNINT(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 DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA )\n CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR)\n CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )\n CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n DO 8998 p = 1, NR\n CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )\n 8998 CONTINUE\n CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n\
*\n CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,\n & LDU, WORK(N+1), LWORK, INFO )\n SCALEM = WORK(N+1)\n NUMRANK = IDNINT(WORK(N+2))\n IF ( NR .LT. N ) THEN\n CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1), LDV )\n CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1), LDV )\n CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV )\n END IF\n\
*\n CALL DORMLQ( '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 DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )\n 8991 CONTINUE\n CALL DLACPY( 'All', N, N, A, LDA, V, LDV )\n\
*\n IF ( TRANSP ) THEN\n CALL DLACPY( '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 DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )\n 1965 CONTINUE\n CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n\
*\n CALL DGEQRF( 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 DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )\n 1967 CONTINUE\n CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n\
*\n CALL DGESVJ( '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 = IDNINT(WORK(N+2))\n\
*\n IF ( NR .LT. M ) THEN\n CALL DLASET( 'A', M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU )\n CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU )\n END IF\n END IF\n\
*\n CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n\
*\n IF ( ROWPIV )\n & CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )\n\
*\n DO 1974 p = 1, N1\n XSC = ONE / DNRM2( M, U(1,p), 1 )\n CALL DSCAL( M, XSC, U(1,p), 1 )\n 1974 CONTINUE\n\
*\n IF ( TRANSP ) THEN\n CALL DLACPY( '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 DGEJSV.\n\
*\n DO 1968 p = 1, NR\n CALL DCOPY( 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 = DSQRT(SMALL)\n DO 2969 q = 1, NR\n TEMP1 = XSC*DABS( V(q,q) )\n DO 2968 p = 1, N\n IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )\n & .OR. ( p .LT. q ) )\n & V(p,q) = DSIGN( 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 DLASET( '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 DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )\n DO 3950 p = 1, NR\n TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)\n CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)\n 3950 CONTINUE\n CALL DPOCON('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 / DSQRT(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. DBLE(N)\n\
* more conservative <=> CONDR1 .LT. DSQRT(DBLE(N))\n\
*\n COND_OK = DSQRT(DBLE(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 DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n\
*\n IF ( L2PERT ) THEN\n XSC = DSQRT(SMALL)/EPSLN\n DO 3959 p = 2, NR\n DO 3958 q = 1, p - 1\n TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n IF ( DABS(V(q,p)) .LE. TEMP1 )\n & V(q,p) = DSIGN( TEMP1, V(q,p) )\n 3958 CONTINUE\n 3959 CONTINUE\n END IF\n\
*\n IF ( NR .NE. N )\n\
* .. save ...\n & CALL DLACPY( '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 DCOPY( 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 DGEQP3\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 DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),\n & WORK(2*N+1), LWORK-2*N, IERR )\n\
** CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n\
** & LWORK-2*N, IERR )\n IF ( L2PERT ) THEN\n XSC = DSQRT(SMALL)\n DO 3969 p = 2, NR\n DO 3968 q = 1, p - 1\n TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n IF ( DABS(V(q,p)) .LE. TEMP1 )\n & V(q,p) = DSIGN( TEMP1, V(q,p) )\n 3968 CONTINUE\n 3969 CONTINUE\n END IF\n\
*\n CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )\n\
*\n IF ( L2PERT ) THEN\n XSC = DSQRT(SMALL)\n DO 8970 p = 2, NR\n DO 8971 q = 1, p - 1\n TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))\n V(p,q) = - DSIGN( TEMP1, V(q,p) )\n 8971 CONTINUE\n 8970 CONTINUE\n ELSE\n CALL DLASET( '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 DGELQF( 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 DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )\n DO 4950 p = 1, NR\n TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR )\n CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )\n 4950 CONTINUE\n CALL DPOCON( '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 / DSQRT(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 DLACPY( '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 = DSQRT(SMALL)\n DO 4968 q = 2, NR\n TEMP1 = XSC * V(q,q)\n DO 4969 p = 1, q - 1\n\
* V(p,q) = - DSIGN( TEMP1, V(q,p) )\n V(p,q) = - DSIGN( TEMP1, V(p,q) )\n 4969 CONTINUE\n 4968 CONTINUE\n ELSE\n CALL DLASET( '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 DGESVJ( '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 = IDNINT(WORK(2*N+N*NR+NR+2))\n DO 3970 p = 1, NR\n CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )\n CALL DSCAL( 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 DGESVJ, premultiplied with the orthogonal matrix\n\
* from the second QR factorization.\n CALL DTRSM( '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 DGESVJ. The Q-factor from the second QR\n\
* factorization is then built in explicitly.\n CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),\n & N,V,LDV)\n IF ( NR .LT. N ) THEN\n CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)\n CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)\n CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)\n END IF\n CALL DORMQR('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 DGESVJ( '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 = IDNINT(WORK(2*N+N*NR+NR+2))\n DO 3870 p = 1, NR\n CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )\n CALL DSCAL( NR, SVA(p), U(1,p), 1 )\n 3870 CONTINUE\n CALL DTRSM('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 DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n CALL DORMQR( '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 DGEJSV completes the task.\n\
* Compute the full SVD of L3 using DGESVJ with explicit\n\
* accumulation of Jacobi rotations.\n CALL DGESVJ( '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 = IDNINT(WORK(2*N+N*NR+NR+2))\n IF ( NR .LT. N ) THEN\n CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n CALL DORMQR( '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 DORMLQ( '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 = DSQRT(DBLE(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 / DNRM2( N, V(1,q), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL DSCAL( 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 DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)\n CALL DLASET('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 DORMQR( '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 = DSQRT(DBLE(M)) * EPSLN\n DO 1973 p = 1, NR\n XSC = ONE / DNRM2( M, U(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL DSCAL( 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 DLASWP( 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 DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )\n IF ( L2PERT ) THEN\n XSC = DSQRT(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)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q))\n 5971 CONTINUE\n 5970 CONTINUE\n ELSE\n CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )\n END IF\n\
*\n CALL DGESVJ( '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 = IDNINT(WORK(N+N*N+2))\n DO 6970 p = 1, N\n CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )\n CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )\n 6970 CONTINUE\n\
*\n CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,\n & ONE, A, LDA, WORK(N+1), N )\n DO 6972 p = 1, N\n CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )\n 6972 CONTINUE\n TEMP1 = DSQRT(DBLE(N))*EPSLN\n DO 6971 p = 1, N\n XSC = ONE / DNRM2( N, V(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL DSCAL( 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 DLASET( 'A', M-N, N, ZERO, ZERO, U(N+1,1), LDU )\n IF ( N .LT. N1 ) THEN\n CALL DLASET( 'A',N, N1-N, ZERO, ZERO, U(1,N+1),LDU )\n CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU )\n END IF\n END IF\n CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n TEMP1 = DSQRT(DBLE(M))*EPSLN\n DO 6973 p = 1, N1\n XSC = ONE / DNRM2( M, U(1,p), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL DSCAL( M, XSC, U(1,p), 1 )\n 6973 CONTINUE\n\
*\n IF ( ROWPIV )\n & CALL DLASWP( 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 DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )\n 7968 CONTINUE\n\
*\n IF ( L2PERT ) THEN\n XSC = DSQRT(SMALL/EPSLN)\n DO 5969 q = 1, NR\n TEMP1 = XSC*DABS( V(q,q) )\n DO 5968 p = 1, N\n IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )\n & .OR. ( p .LT. q ) )\n & V(p,q) = DSIGN( 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 DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )\n END IF\n\n CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),\n & LWORK-2*N, IERR )\n CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )\n\
*\n DO 7969 p = 1, NR\n CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )\n 7969 CONTINUE\n\n IF ( L2PERT ) THEN\n XSC = DSQRT(SMALL/EPSLN)\n DO 9970 q = 2, NR\n DO 9971 p = 1, q - 1\n TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q)))\n U(p,q) = - DSIGN( TEMP1, U(q,p) )\n 9971 CONTINUE\n 9970 CONTINUE\n ELSE\n CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )\n END IF\n\n CALL DGESVJ( 'G', '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 = IDNINT(WORK(2*N+N*NR+2))\n\n IF ( NR .LT. N ) THEN\n CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )\n CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )\n CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )\n END IF\n\n CALL DORMQR( '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 = DSQRT(DBLE(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 / DNRM2( N, V(1,q), 1 )\n IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )\n & CALL DSCAL( 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 DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )\n IF ( NR .LT. N1 ) THEN\n CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),LDU )\n CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU )\n END IF\n END IF\n\
*\n CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,\n & LDU, WORK(N+1), LWORK-N, IERR )\n\
*\n IF ( ROWPIV )\n & CALL DLASWP( 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 DSWAP( 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 DLASCL( '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 DGEJSV\n\
* ..\n END\n\
*\n"
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