NAG Library Routine Document
F08MSF (ZBDSQR)
1 Purpose
F08MSF (ZBDSQR) computes the singular value decomposition of a complex general matrix which has been reduced to bidiagonal form.
2 Specification
| SUBROUTINE F08MSF ( |
UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C, LDC, WORK, INFO) |
| INTEGER |
N, NCVT, NRU, NCC, LDVT, LDU, LDC, INFO |
| REAL (KIND=nag_wp) |
D(*), E(*), WORK(*) |
| COMPLEX (KIND=nag_wp) |
VT(LDVT,*), U(LDU,*), C(LDC,*) |
| CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name zbdsqr.
3 Description
F08MSF (ZBDSQR) computes the singular values and, optionally, the left or right singular vectors of a real upper or lower bidiagonal matrix
B. In other words, it can compute the singular value decomposition (SVD) of
B as
Here
Σ is a diagonal matrix with real diagonal elements
σi (the singular values of
B), such that
U is an orthogonal matrix whose columns are the left singular vectors
ui;
V is an orthogonal matrix whose rows are the right singular vectors
vi. Thus
To compute
U and/or
VT, the arrays
U and/or
VT must be initialized to the unit matrix before F08MSF (ZBDSQR) is called.
The routine stores the real orthogonal matrices
U and
VT in complex arrays
U and
VT, so that it may also be used to compute the SVD of a complex general matrix
A which has been reduced to bidiagonal form by a unitary transformation:
A=QBPH. If
A is
m by
n with
m≥n, then
Q is
m by
n and
PH is
n by
n; if
A is
n by
p with
n<p, then
Q is
n by
n and
PH is
n by
p. In this case, the matrices
Q and/or
PH must be formed explicitly by
F08KTF (ZUNGBR) and passed to F08MSF (ZBDSQR) in the arrays
U and/or
VT respectively.
F08MSF (ZBDSQR) also has the capability of forming UHC, where C is an arbitrary complex matrix; this is needed when using the SVD to solve linear least squares problems.
F08MSF (ZBDSQR) uses two different algorithms. If any singular vectors are required (i.e., if
NCVT>0 or
NRU>0 or
NCC>0), the bidiagonal
QR algorithm is used, switching between zero-shift and implicitly shifted forms to preserve the accuracy of small singular values, and switching between
QR and
QL variants in order to handle graded matrices effectively (see
Demmel and Kahan (1990)). If only singular values are required (i.e., if
NCVT=NRU=NCC=0), they are computed by the differential qd algorithm (see
Fernando and Parlett (1994)), which is faster and can achieve even greater accuracy.
The singular vectors are normalized so that ui=vi=1, but are determined only to within a complex factor of absolute value 1.
4 References
Demmel J W and Kahan W (1990) Accurate singular values of bidiagonal matrices
SIAM J. Sci. Statist. Comput. 11 873–912
Fernando K V and Parlett B N (1994) Accurate singular values and differential qd algorithms
Numer. Math. 67 191–229
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: UPLO – CHARACTER(1)Input
-
On entry: indicates whether
B is an upper or lower bidiagonal matrix.
- UPLO='U'
- B is an upper bidiagonal matrix.
- UPLO='L'
- B is a lower bidiagonal matrix.
Constraint:
UPLO='U' or 'L'.
- 2: N – INTEGERInput
-
On entry:
n, the order of the matrix B.
Constraint:
N≥0.
- 3: NCVT – INTEGERInput
-
On entry: ncvt, the number of columns of the matrix
VHNCVT=0 of right singular vectors. Set NCVT=0 if no right singular vectors are required.
Constraint:
NCVT≥0.
- 4: NRU – INTEGERInput
-
On entry:
nru, the number of rows of the matrix U of left singular vectors. Set NRU=0 if no left singular vectors are required.
Constraint:
NRU≥0.
- 5: NCC – INTEGERInput
-
On entry: ncc, the number of columns of the matrix C. Set NCC=0 if no matrix C is supplied.
Constraint:
NCC≥0.
- 6: D(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the diagonal elements of the bidiagonal matrix B.
On exit: the singular values in decreasing order of magnitude, unless
INFO>0 (in which case see
Section 6).
- 7: E(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On entry: the off-diagonal elements of the bidiagonal matrix B.
On exit:
E is overwritten, but if
INFO>0 see
Section 6.
- 8: VT(LDVT,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
VT
must be at least
max1,NCVT.
On entry: if
NCVT>0,
VT must contain an
n by
ncvt matrix. If the right singular vectors of
B are required,
ncvt=n and
VT must contain the unit matrix; if the right singular vectors of
A are required,
VT must contain the unitary
matrix
PH
returned by
F08KTF (ZUNGBR)
with
VECT='P' .
On exit: the
n by
ncvt matrix
VH
or
VH
of right singular vectors, stored by rows.
If
NCVT=0,
VT is not referenced.
- 9: LDVT – INTEGERInput
-
On entry: the first dimension of the array
VT as declared in the (sub)program from which F08MSF (ZBDSQR) is called.
Constraints:
- if NCVT>0, LDVT≥max1,N;
- otherwise LDVT≥1.
- 10: U(LDU,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
U
must be at least
max1,N.
On entry: if
NRU>0,
U must contain an
nru by
n matrix. If the left singular vectors of
B are required,
nru=n and
U must contain the unit matrix; if the left singular vectors of
A are required,
U must contain the unitary
matrix
Q returned by
F08KTF (ZUNGBR)
with
VECT='Q' .
On exit: the
nru by
n matrix
U or
QU of left singular vectors, stored as columns of the matrix.
If
NRU=0,
U is not referenced.
- 11: LDU – INTEGERInput
-
On entry: the first dimension of the array
U as declared in the (sub)program from which F08MSF (ZBDSQR) is called.
Constraint:
LDU≥max1,NRU.
- 12: C(LDC,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
C
must be at least
max1,NCC.
On entry: the n by ncc matrix C if NCC>0.
On exit:
C is overwritten by the matrix
UHC. If
NCC=0,
C is not referenced.
- 13: LDC – INTEGERInput
-
On entry: the first dimension of the array
C as declared in the (sub)program from which F08MSF (ZBDSQR) is called.
Constraints:
- if NCC>0, LDC≥max1,N;
- otherwise LDC≥1.
- 14: WORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
WORK
must be at least
max1,2×N if
NCVT=NRU=NCC=0, and at least
max1,4×N otherwise.
- 15: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
The algorithm failed to converge and
INFO specifies how many off-diagonals did not converge. In this case,
D and
E contain on exit the diagonal and off-diagonal elements, respectively, of a bidiagonal matrix orthogonally equivalent to
B.
7 Accuracy
Each singular value and singular vector is computed to high relative accuracy. However, the reduction to bidiagonal form (prior to calling the routine) may exclude the possibility of obtaining high relative accuracy in the small singular values of the original matrix if its singular values vary widely in magnitude.
If
σi is an exact singular value of
B and
σ~i is the corresponding computed value, then
where
pm,n is a modestly increasing function of
m and
n, and
ε is the
machine precision. If only singular values are computed, they are computed more accurately (i.e., the function
pm,n is smaller), than when some singular vectors are also computed.
If
ui is an exact left singular vector of
B, and
u~i is the corresponding computed left singular vector, then the angle
θu~i,ui between them is bounded as follows:
where
relgapi is the relative gap between
σi and the other singular values, defined by
A similar error bound holds for the right singular vectors.
8 Further Comments
The total number of real floating point operations is roughly proportional to n2 if only the singular values are computed. About 12n2×nru additional operations are required to compute the left singular vectors and about 12n2×ncvt to compute the right singular vectors. The operations to compute the singular values must all be performed in scalar mode; the additional operations to compute the singular vectors can be vectorized and on some machines may be performed much faster.
The real analogue of this routine is
F08MEF (DBDSQR).
9 Example
See
Section 9 in F08KTF (ZUNGBR), which illustrates the use of the routine to compute the singular value decomposition of a general matrix.