NAME
PDGESVD - compute the singular value decomposition (SVD) of an M-by-N
matrix A, optionally computing the left and/or right singular vectors
SYNOPSIS
SUBROUTINE PDGESVD( JOBU, JOBVT, M, N, A, IA, JA, DESCA, S, U, IU, JU,
DESCU, VT, IVT, JVT, DESCVT, WORK, LWORK, INFO )
CHARACTER JOBU, JOBVT
INTEGER IA, INFO, IU, IVT, JA, JU, JVT, LWORK, M, N
INTEGER DESCA( * ), DESCU( * ), DESCVT( * )
DOUBLE PRECISION A( * ), S( * ), U( * ), VT( * ), WORK( *
)
PURPOSE
PDGESVD computes the singular value decomposition (SVD) of an M-by-N
matrix A, optionally computing the left and/or right singular vectors.
The SVD is written as
A = U * SIGMA * transpose(V)
where SIGMA is an M-by-N matrix which is zero except for its min(M,N)
diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N
orthogonal matrix. The diagonal elements of SIGMA are the singular
values of A and the columns of U and V are the corresponding right and
left singular vectors, respectively. The singular values are returned
in array S in decreasing order and only the first min(M,N) columns of U
and rows of VT = V**T are computed.
Notes
=====
Each global data object is described by an associated description
vector. This vector stores the information required to establish the
mapping between an object element and its corresponding process and
memory location.
Let A be a generic term for any 2D block cyclicly distributed array.
Such a global array has an associated description vector DESCA. In the
following comments, the character _ should be read as "of the global
array".
NOTATION STORED IN EXPLANATION
--------------- -------------- --------------------------------------
DTYPE_A(global) DESCA( DTYPE_ )The descriptor type. In this case,
DTYPE_A = 1.
CTXT_A (global) DESCA( CTXT_ ) The BLACS context handle, indicating
the BLACS process grid A is distribu-
ted over. The context itself is glo-
bal, but the handle (the integer
value) may vary.
M_A (global) DESCA( M_ ) The number of rows in the global
array A.
N_A (global) DESCA( N_ ) The number of columns in the global
array A.
MB_A (global) DESCA( MB_ ) The blocking factor used to distribute
the rows of the array.
NB_A (global) DESCA( NB_ ) The blocking factor used to distribute
the columns of the array.
RSRC_A (global) DESCA( RSRC_ ) The process row over which the first
row of the array A is distributed.
CSRC_A (global) DESCA( CSRC_ ) The process column over which the
first column of the array A is
distributed.
LLD_A (local) DESCA( LLD_ ) The leading dimension of the local
array. LLD_A >= MAX(1,LOCr(M_A)).
Let K be the number of rows or columns of a distributed matrix, and
assume that its process grid has dimension p x q. LOCr( K ) denotes the
number of elements of K that a process would receive if K were
distributed over the p processes of its process column. Similarly,
LOCc( K ) denotes the number of elements of K that a process would
receive if K were distributed over the q processes of its process row.
The values of LOCr() and LOCc() may be determined via a call to the
ScaLAPACK tool function, NUMROC:
LOCr( M ) = NUMROC( M, MB_A, MYROW, RSRC_A, NPROW ),
LOCc( N ) = NUMROC( N, NB_A, MYCOL, CSRC_A, NPCOL ). An upper
bound for these quantities may be computed by:
LOCr( M ) <= ceil( ceil(M/MB_A)/NPROW )*MB_A
LOCc( N ) <= ceil( ceil(N/NB_A)/NPCOL )*NB_A
ARGUMENTS
MP = number of local rows in A and U NQ = number of local columns in A
and VT SIZE = min( M, N ) SIZEQ = number of local columns in U SIZEP =
number of local rows in VT
JOBU (global input) CHARACTER*1
Specifies options for computing all or part of the matrix U:
= ’V’: the first SIZE columns of U (the left singular vectors)
are returned in the array U; = ’N’: no columns of U (no left
singular vectors) are computed.
JOBVT (global input) CHARACTER*1
Specifies options for computing all or part of the matrix V**T:
= ’V’: the first SIZE rows of V**T (the right singular
vectors) are returned in the array VT; = ’N’: no rows of V**T
(no right singular vectors) are computed.
M (global input) INTEGER
The number of rows of the input matrix A. M >= 0.
N (global input) INTEGER
The number of columns of the input matrix A. N >= 0.
A (local input/workspace) block cyclic DOUBLE PRECISION array,
global dimension (M, N), local dimension (MP, NQ) On exit, the
contents of A are destroyed.
IA (global input) INTEGER
The row index in the global array A indicating the first row of
sub( A ).
JA (global input) INTEGER
The column index in the global array A indicating the first
column of sub( A ).
DESCA (global input) INTEGER array of dimension DLEN_
The array descriptor for the distributed matrix A.
S (global output) DOUBLE PRECISION array, dimension SIZE
The singular values of A, sorted so that S(i) >= S(i+1).
U (local output) DOUBLE PRECISION array, local dimension
(MP, SIZEQ), global dimension (M, SIZE) if JOBU = ’V’, U
contains the first min(m,n) columns of U if JOBU = ’N’, U is
not referenced.
IU (global input) INTEGER
The row index in the global array U indicating the first row of
sub( U ).
JU (global input) INTEGER
The column index in the global array U indicating the first
column of sub( U ).
DESCU (global input) INTEGER array of dimension DLEN_
The array descriptor for the distributed matrix U.
VT (local output) DOUBLE PRECISION array, local dimension
(SIZEP, NQ), global dimension (SIZE, N). If JOBVT = ’V’, VT
contains the first SIZE rows of V**T. If JOBVT = ’N’, VT is not
referenced.
IVT (global input) INTEGER
The row index in the global array VT indicating the first row
of sub( VT ).
JVT (global input) INTEGER
The column index in the global array VT indicating the first
column of sub( VT ).
DESCVT (global input) INTEGER array of dimension DLEN_
The array descriptor for the distributed matrix VT.
WORK (local workspace/output) DOUBLE PRECISION array, dimension
(LWORK) On exit, if INFO = 0, WORK(1) returns the optimal
LWORK;
LWORK (local input) INTEGER
The dimension of the array WORK.
LWORK > 2 + 6*SIZEB + MAX(WATOBD, WBDTOSVD),
where SIZEB = MAX(M,N), and WATOBD and WBDTOSVD refer,
respectively, to the workspace required to bidiagonalize the
matrix A and to go from the bidiagonal matrix to the singular
value decomposition U*S*VT.
For WATOBD, the following holds:
WATOBD = MAX(MAX(WPDLANGE,WPDGEBRD),
MAX(WPDLARED2D,WPDLARED1D)),
where WPDLANGE, WPDLARED1D, WPDLARED2D, WPDGEBRD are the
workspaces required respectively for the subprograms PDLANGE,
PDLARED1D, PDLARED2D, PDGEBRD. Using the standard notation
MP = NUMROC( M, MB, MYROW, DESCA( CTXT_ ), NPROW), NQ = NUMROC(
N, NB, MYCOL, DESCA( LLD_ ), NPCOL),
the workspaces required for the above subprograms are
WPDLANGE = MP, WPDLARED1D = NQ0, WPDLARED2D = MP0, WPDGEBRD =
NB*(MP + NQ + 1) + NQ,
where NQ0 and MP0 refer, respectively, to the values obtained
at MYCOL = 0 and MYROW = 0. In general, the upper limit for the
workspace is given by a workspace required on processor (0,0):
WATOBD <= NB*(MP0 + NQ0 + 1) + NQ0.
In case of a homogeneous process grid this upper limit can be
used as an estimate of the minimum workspace for every
processor.
For WBDTOSVD, the following holds:
WBDTOSVD = SIZE*(WANTU*NRU + WANTVT*NCVT) + MAX(WDBDSQR,
MAX(WANTU*WPDORMBRQLN, WANTVT*WPDORMBRPRT)),
where
1, if left(right) singular vectors are wanted WANTU(WANTVT) = 0,
otherwise
and WDBDSQR, WPDORMBRQLN and WPDORMBRPRT refer respectively to the
workspace required for the subprograms DBDSQR, PDORMBR(QLN), and
PDORMBR(PRT), where QLN and PRT are the values of the arguments VECT,
SIDE, and TRANS in the call to PDORMBR. NRU is equal to the local number
of rows of the matrix U when distributed 1-dimensional "column" of
processes. Analogously, NCVT is equal to the local number of columns of
the matrix VT when distributed across 1-dimensional "row" of processes.
Calling the LAPACK procedure DBDSQR requires
WDBDSQR = MAX(1, 2*SIZE + (2*SIZE - 4)*MAX(WANTU, WANTVT))
on every processor. Finally,
WPDORMBRQLN = MAX( (NB*(NB-1))/2, (SIZEQ+MP)*NB)+NB*NB, WPDORMBRPRT =
MAX( (MB*(MB-1))/2, (SIZEP+NQ)*MB )+MB*MB,
If LIWORK = -1, then LIWORK is global input and a workspace query is
assumed; the routine only calculates the minimum size for the work
array. The required workspace is returned as the first element of WORK
and no error message is issued by PXERBLA.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if SBDSQR did not converge If INFO = MIN(M,N) + 1, then
PDGESVD has detected heterogeneity by finding that eigenvalues
were not identical across the process grid. In this case, the
accuracy of the results from PDGESVD cannot be guaranteed.