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template<typename T>
void lasd1(integer *nl, integer *nr, integer *sqre, T *d, T *alpha, T *beta, T *u, integer *ldu, T *vt, integer *ldvt, integer *idxq, integer *iwork, T *work, integer *info)# LASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
Purpose:
LASD1 computes the SVD of an upper bidiagonal N-by-M matrix B, where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0. A related subroutine SLASD7 handles the case in which the singular values (and the singular vectors in factored form) are desired. SLASD1 computes the SVD as follows: (D1(in) 0 0 0) B = U(in) * ( Z1**T a Z2**T b) * VT(in) ( 0 0 D2(in) 0) = U(out) * (D(out) 0) * VT(out) where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and the entry b is empty if SQRE = 0. The left singular vectors of the original matrix are stored in U, and the transpose of the right singular vectors are stored in VT, and the singular values are in D. The algorithm consists of three stages: The first stage consists of deflating the size of the problem when there are multiple singular values or when there are zeros in the Z vector. For each such occurrence the dimension of the secular equation problem is reduced by one. This stage is performed by the routine SLASD2. The second stage consists of calculating the updated singular values. This is done by finding the square roots of the roots of the secular equation via the routine SLASD4 (as called by SLASD3). This routine also calculates the singular vectors of the current problem. The final stage consists of computing the updated singular vectors directly using the updated singular values. The singular vectors for the current problem are multiplied with the singular vectors from the overall problem.
- Parameters:
NL – [in]
NL is INTEGER
The row dimension of the upper block. NL >= 1.
NR – [in]
NR is INTEGER
The row dimension of the lower block. NR >= 1.
SQRE – [in]
SQRE is INTEGER
= 0: the lower block is an NR-by-NR square matrix.
= 1: the lower block is an NR-by-(NR+1) rectangular matrix.
The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.
D – [inout]
D is REAL array, dimension (NL+NR+1).
N = NL+NR+1
On entry D(1:NL,1:NL) contains the singular values of the upper block; and D(NL+2:N) contains the singular values of the lower block. On exit D(1:N) contains the singular values of the modified matrix.ALPHA – [inout]
ALPHA is REAL
Contains the diagonal element associated with the added row.
BETA – [inout]
BETA is REAL
Contains the off-diagonal element associated with the added row.
U – [inout]
U is REAL array, dimension (LDU,N)
On entry U(1:NL, 1:NL) contains the left singular vectors of the upper block; U(NL+2:N, NL+2:N) contains the left singular vectors of the lower block. On exit U contains the left singular vectors of the bidiagonal matrix.
LDU – [in]
LDU is INTEGER
The leading dimension of the array U. LDU >= fla_max(1, N).
VT – [inout]
VT is REAL array, dimension (LDVT,M)
where M = N + SQRE.
On entry VT(1:NL+1, 1:NL+1)**T contains the right singular vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains the right singular vectors of the lower block. On exit VT**T contains the right singular vectors of the bidiagonal matrix.LDVT – [in]
LDVT is INTEGER
The leading dimension of the array VT. LDVT >= fla_max(1, M).
IDXQ – [inout]
IDXQ is INTEGER array, dimension (N)
This contains the permutation which will reintegrate the subproblem just solved back into sorted order, i.e. D(IDXQ(I = 1, N)) will be in ascending order.
IWORK – [out] IWORK is INTEGER array, dimension (4*N)
WORK – [out] WORK is REAL array, dimension (3*M**2+2*M)
INFO – [out]
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = 1, a singular value did not converge