Computes the singular value decomposition of a general matrix, optionally computing the left and/or right singular vectors.
call psgesvd(jobu, jobvt, m, n, a, ia, ja, desca, s, u, iu, ju, descu, vt, ivt, jvt, descvt, work, lwork, info)
call pdgesvd(jobu, jobvt, m, n, a, ia, ja, desca, s, u, iu, ju, descu, vt, ivt, jvt, descvt, work, lwork, info)
call pcgesvd(jobu, jobvt, m, n, a, ia, ja, desca, s, u, iu, ju, descu, vt, ivt, jvt, descvt, work, lwork, rwork, info)
call pzgesvd(jobu, jobvt, m, n, a, ia, ja, desca, s, u, iu, ju, descu, vt, ivt, jvt, descvt, work, lwork, rwork, info)
The p?gesvd routine 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
A = U*Σ*VT,
where Σ is an m-by-n matrix that 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 Σ 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 = VT are computed.
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
(global). CHARACTER*1. Specifies options for computing all or part of the matrix U.
If jobu = 'V', the first size columns of U (the left singular vectors) are returned in the array u;
If jobu ='N', no columns of U (no left singular vectors)are computed.
(global) CHARACTER*1.
Specifies options for computing all or part of the matrix VT.
If jobvt = 'V', the first size rows of VT (the right singular vectors) are returned in the array vt;
If jobvt = 'N', no rows of VT(no right singular vectors) are computed.
(global) INTEGER. The number of rows of the matrix A (m ≥ 0).
(global) INTEGER. The number of columns in A (n ≥ 0).
(local). REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
Block cyclic array, global dimension (m, n), local dimension (mp, nq).
work(lwork) is a workspace array.
(global) INTEGER. The row and column indices in the global array a indicating the first row and the first column of the submatrix A, respectively.
(global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix A.
(global) INTEGER. The row and column indices in the global array a indicating the first row and the first column of the submatrix U, respectively.
(global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix U.
(global) INTEGER. The row and column indices in the global array vt indicating the first row and the first column of the submatrix VT, respectively.
(global and local) INTEGER array, dimension (dlen_). The array descriptor for the distributed matrix VT.
(local). REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
Workspace array, dimension (lwork)
(local) 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(wp?lange,wp?gebrd), max(wp?lared2d, wp?lared1d)),
where wp?lange, wp?lared1d, wp?lared2d, wp?gebrd are the workspaces required respectively for the subprograms p?lange, p?lared1d, p?lared2d, p?gebrd. 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
wp?lange = mp,
wp?lared1d = nq0,
wp?lared2d = mp0,
wp?gebrd = 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(w?bdsqr, max(wantu*wp?ormbrqln, wantvt*wp?ormbrprt)),
where
wantu(wantvt) = 1, if left/right singular vectors are wanted, and wantu(wantvt) = 0, otherwise. w?bdsqr, wp?ormbrqln, and wp?ormbrprt refer respectively to the workspace required for the subprograms ?bdsqr, p?ormbr(qln), and p?ormbr(prt), where qln and prt are the values of the arguments vect, side, and trans in the call to p?ormbr. 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 ?bdsqr requires
w?bdsqr = max(1, 2*size + (2*size - 4)* max(wantu, wantvt))
on every processor. Finally,
wp?ormbrqln = max((nb*(nb-1))/2, (sizeq+mp)*nb)+nb*nb,
wp?ormbrprt = max((mb*(mb-1))/2, (sizep+nq)*mb)+mb*mb,
If lwork = -1, then lwork 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.
REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
Workspace array, dimension (1 + 4*sizeb)
On exit, the contents of a are destroyed.
(global). REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
Array, DIMENSION (size).
Contains the singular values of A sorted so that s(i) ≥ s(i+1).
(local). REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
local dimension (mp, sizeq), global dimension (m, size)
If jobu = 'V', u contains the first min(m, n) columns of U.
If jobu = 'N' or 'O', u is not referenced.
(local). REAL for psgesvd
DOUBLE PRECISION for pdgesvd
COMPLEX for pcgesvd
COMPLEX*16 for pzgesvd
local dimension (sizep, nq), global dimension (size, n)
If jobvt = 'V', vt contains the first size rows of VTif jobu = 'N', vt is not referenced.
On exit, if info = 0, then work(1) returns the required minimal size of lwork.
On exit, if info = 0, then rwork(1) returns the required size of rwork.
(global) INTEGER.
If info = 0, the execution is successful.
If info < 0, If info = -i, the ith parameter had an illegal value.
If info > 0 i, then if ?bdsqr did not converge,
If info = min(m,n) + 1, then p?gesvd has detected heterogeneity by finding that eigenvalues were not identical across the process grid. In this case, the accuracy of the results from p?gesvd cannot be guaranteed.
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