NAG Library Routine Document
F08MEF (DBDSQR)
1 Purpose
F08MEF (DBDSQR) computes the singular value decomposition of a real upper or lower bidiagonal matrix, or of a real general matrix which has been reduced to bidiagonal form.
2 Specification
SUBROUTINE F08MEF ( |
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(*), VT(LDVT,*), U(LDU,*), C(LDC,*), WORK(*) |
CHARACTER(1) |
UPLO |
|
The routine may be called by its
LAPACK
name dbdsqr.
3 Description
F08MEF (DBDSQR) 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 F08MEF (DBDSQR) is called.
The routine may also be used to compute the SVD of a real general matrix
A which has been reduced to bidiagonal form by an orthogonal transformation:
A=QBPT. If
A is
m by
n with
m≥n, then
Q is
m by
n and
PT is
n by
n; if
A is
n by
p with
n<p, then
Q is
n by
n and
PT is
n by
p. In this case, the matrices
Q and/or
PT must be formed explicitly by
F08KFF (DORGBR) and passed to F08MEF (DBDSQR) in the arrays
U and/or
VT respectively.
F08MEF (DBDSQR) also has the capability of forming UTC, where C is an arbitrary real matrix; this is needed when using the SVD to solve linear least squares problems.
F08MEF (DBDSQR) 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 factor ±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 VT 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,*) – REAL (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 orthogonal matrix
PT returned by
F08KFF (DORGBR) with
VECT='P' .
On exit: the
n by
ncvt matrix
VT or
VTPT 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 F08MEF (DBDSQR) is called.
Constraints:
- if NCVT>0, LDVT≥ max1,N ;
- otherwise LDVT≥1.
- 10: U(LDU,*) – REAL (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 orthogonal matrix
Q returned by
F08KFF (DORGBR) 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 F08MEF (DBDSQR) is called.
Constraint:
LDU≥ max1,NRU .
- 12: C(LDC,*) – REAL (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
UTC. 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 F08MEF (DBDSQR) 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=0 and
NRU=0 and
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 the corresponding 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 floating point operations is roughly proportional to n2 if only the singular values are computed. About 6n2×nru additional operations are required to compute the left singular vectors and about 6n2×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 complex analogue of this routine is
F08MSF (ZBDSQR).
9 Example
This example computes the singular value decomposition of the upper bidiagonal matrix
B, where
See also the example for
F08KFF (DORGBR), which illustrates the use of the routine to compute the singular value decomposition of a general matrix.
9.1 Program Text
Program Text (f08mefe.f90)
9.2 Program Data
Program Data (f08mefe.d)
9.3 Program Results
Program Results (f08mefe.r)