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---
:name: zgelsd
:md5sum: 3cda6a4e059bdc95d6c1841204c3d144
:category: :subroutine
:arguments:
- m:
:type: integer
:intent: input
- n:
:type: integer
:intent: input
- nrhs:
:type: integer
:intent: input
- a:
:type: doublecomplex
:intent: input
:dims:
- lda
- n
- lda:
:type: integer
:intent: input
- b:
:type: doublecomplex
:intent: input/output
:dims:
- m
- nrhs
:outdims:
- n
- nrhs
- ldb:
:type: integer
:intent: input
- s:
:type: doublereal
:intent: output
:dims:
- MIN(m,n)
- rcond:
:type: doublereal
:intent: input
- rank:
:type: integer
:intent: output
- work:
:type: doublecomplex
:intent: output
:dims:
- MAX(1,lwork)
- lwork:
:type: integer
:intent: input
:option: true
:default: "m>=n ? 2*n+n*nrhs : 2*m+m*nrhs"
- rwork:
:type: doublereal
:intent: workspace
:dims:
- MAX(1,lrwork)
- iwork:
:type: integer
:intent: workspace
:dims:
- MAX(1,liwork)
- info:
:type: integer
:intent: output
:substitutions:
m: lda
ldb: MAX(m,n)
c__9: "9"
c__0: "0"
liwork: MAX(1,3*(MIN(m,n))*nlvl+11*(MIN(m,n)))
lrwork: "m>=n ? 10*n+2*n*smlsiz+8*n*nlvl+3*smlsiz*nrhs+(smlsiz+1)*(smlsiz+1) : 10*m+2*m*smlsiz+8*m*nlvl+2*smlsiz*nrhs+(smlsiz+1)*(smlsiz+1)"
nlvl: MAX(0,(int)(log(1.0*MIN(m,n)/(smlsiz+1))/log(2.0)))
smlsiz: ilaenv_(&c__9,"ZGELSD"," ",&c__0,&c__0,&c__0,&c__0)
:fortran_help: " SUBROUTINE ZGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, IWORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* ZGELSD computes the minimum-norm solution to a real linear least\n\
* squares problem:\n\
* minimize 2-norm(| b - A*x |)\n\
* using the singular value decomposition (SVD) of A. A is an M-by-N\n\
* matrix which may be rank-deficient.\n\
*\n\
* Several right hand side vectors b and solution vectors x can be\n\
* handled in a single call; they are stored as the columns of the\n\
* M-by-NRHS right hand side matrix B and the N-by-NRHS solution\n\
* matrix X.\n\
*\n\
* The problem is solved in three steps:\n\
* (1) Reduce the coefficient matrix A to bidiagonal form with\n\
* Householder transformations, reducing the original problem\n\
* into a \"bidiagonal least squares problem\" (BLS)\n\
* (2) Solve the BLS using a divide and conquer approach.\n\
* (3) Apply back all the Householder transformations to solve\n\
* the original least squares problem.\n\
*\n\
* The effective rank of A is determined by treating as zero those\n\
* singular values which are less than RCOND times the largest singular\n\
* value.\n\
*\n\
* The divide and conquer algorithm makes very mild assumptions about\n\
* floating point arithmetic. It will work on machines with a guard\n\
* digit in add/subtract, or on those binary machines without guard\n\
* digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or\n\
* Cray-2. It could conceivably fail on hexadecimal or decimal machines\n\
* without guard digits, but we know of none.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* M (input) INTEGER\n\
* The number of rows of the matrix A. M >= 0.\n\
*\n\
* N (input) INTEGER\n\
* The number of columns of the matrix A. N >= 0.\n\
*\n\
* NRHS (input) INTEGER\n\
* The number of right hand sides, i.e., the number of columns\n\
* of the matrices B and X. NRHS >= 0.\n\
*\n\
* A (input) COMPLEX*16 array, dimension (LDA,N)\n\
* On entry, the M-by-N matrix A.\n\
* On exit, A has been destroyed.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,M).\n\
*\n\
* B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)\n\
* On entry, the M-by-NRHS right hand side matrix B.\n\
* On exit, B is overwritten by the N-by-NRHS solution matrix X.\n\
* If m >= n and RANK = n, the residual sum-of-squares for\n\
* the solution in the i-th column is given by the sum of\n\
* squares of the modulus of elements n+1:m in that column.\n\
*\n\
* LDB (input) INTEGER\n\
* The leading dimension of the array B. LDB >= max(1,M,N).\n\
*\n\
* S (output) DOUBLE PRECISION array, dimension (min(M,N))\n\
* The singular values of A in decreasing order.\n\
* The condition number of A in the 2-norm = S(1)/S(min(m,n)).\n\
*\n\
* RCOND (input) DOUBLE PRECISION\n\
* RCOND is used to determine the effective rank of A.\n\
* Singular values S(i) <= RCOND*S(1) are treated as zero.\n\
* If RCOND < 0, machine precision is used instead.\n\
*\n\
* RANK (output) INTEGER\n\
* The effective rank of A, i.e., the number of singular values\n\
* which are greater than RCOND*S(1).\n\
*\n\
* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))\n\
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.\n\
*\n\
* LWORK (input) INTEGER\n\
* The dimension of the array WORK. LWORK must be at least 1.\n\
* The exact minimum amount of workspace needed depends on M,\n\
* N and NRHS. As long as LWORK is at least\n\
* 2*N + N*NRHS\n\
* if M is greater than or equal to N or\n\
* 2*M + M*NRHS\n\
* if M is less than N, the code will execute correctly.\n\
* For good performance, LWORK should generally be larger.\n\
*\n\
* If LWORK = -1, then a workspace query is assumed; the routine\n\
* only calculates the optimal size of the array WORK and the\n\
* minimum sizes of the arrays RWORK and IWORK, and returns\n\
* these values as the first entries of the WORK, RWORK and\n\
* IWORK arrays, and no error message related to LWORK is issued\n\
* by XERBLA.\n\
*\n\
* RWORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LRWORK))\n\
* LRWORK >=\n\
* 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +\n\
* MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )\n\
* if M is greater than or equal to N or\n\
* 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +\n\
* MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )\n\
* if M is less than N, the code will execute correctly.\n\
* SMLSIZ is returned by ILAENV and is equal to the maximum\n\
* size of the subproblems at the bottom of the computation\n\
* tree (usually about 25), and\n\
* NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )\n\
* On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))\n\
* LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),\n\
* where MINMN = MIN( M,N ).\n\
* On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit\n\
* < 0: if INFO = -i, the i-th argument had an illegal value.\n\
* > 0: the algorithm for computing the SVD failed to converge;\n\
* if INFO = i, i off-diagonal elements of an intermediate\n\
* bidiagonal form did not converge to zero.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* Based on contributions by\n\
* Ming Gu and Ren-Cang Li, Computer Science Division, University of\n\
* California at Berkeley, USA\n\
* Osni Marques, LBNL/NERSC, USA\n\
*\n\
* =====================================================================\n\
*\n"
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