NAG Library Routine Document
F08MDF (DBDSDC)
1 Purpose
F08MDF (DBDSDC) computes the singular values and, optionally, the left and right singular vectors of a real n by n (upper or lower) bidiagonal matrix B.
2 Specification
| SUBROUTINE F08MDF ( |
UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ, WORK, IWORK, INFO) |
| INTEGER |
N, LDU, LDVT, IQ(*), IWORK(8*N), INFO |
| REAL (KIND=nag_wp) |
D(*), E(*), U(LDU,*), VT(LDVT,*), Q(*), WORK(*) |
| CHARACTER(1) |
UPLO, COMPQ |
|
The routine may be called by its
LAPACK
name dbdsdc.
3 Description
F08MDF (DBDSDC) computes the singular value decomposition (SVD) of the (upper or lower) bidiagonal matrix
B as
where
S is a diagonal matrix with non-negative diagonal elements
sii=si, such that
and
U and
V are orthogonal matrices. The diagonal elements of
S are the singular values of
B and the columns of
U and
V are respectively the corresponding left and right singular vectors of
B.
When only singular values are required the routine uses the QR algorithm, but when singular vectors are required a divide and conquer method is used. The singular values can optionally be returned in compact form, although currently no routine is available to apply U or V when stored in compact form.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lugGolub 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 upper or lower bidiagonal.
- UPLO='U'
- B is upper bidiagonal.
- UPLO='L'
- B is lower bidiagonal.
Constraint:
UPLO='U' or 'L'.
- 2: COMPQ – CHARACTER(1)Input
-
On entry: specifies whether singular vectors are to be computed.
- COMPQ='N'
- Compute singular values only.
- COMPQ='P'
- Compute singular values and compute singular vectors in compact form.
- COMPQ='I'
- Compute singular values and singular vectors.
Constraint:
COMPQ='N', 'P' or 'I'.
- 3: N – INTEGERInput
-
On entry: n, the order of the matrix B.
Constraint:
N≥0.
- 4: D(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
D
must be at least
max1,N.
On entry: the n diagonal elements of the bidiagonal matrix B.
On exit: if INFO=0, the singular values of B.
- 5: E(*) – REAL (KIND=nag_wp) arrayInput/Output
-
Note: the dimension of the array
E
must be at least
max1,N-1.
On entry: the n-1 off-diagonal elements of the bidiagonal matrix B.
On exit: the contents of
E are destroyed.
- 6: U(LDU,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
U
must be at least
max1,N if
COMPQ='I', and at least
1 otherwise.
On exit: if
COMPQ='I', then if
INFO=0,
U contains the left singular vectors of the bidiagonal matrix
B.
If
COMPQ≠'I',
U is not referenced.
- 7: LDU – INTEGERInput
-
On entry: the first dimension of the array
U as declared in the (sub)program from which F08MDF (DBDSDC) is called.
Constraints:
- if COMPQ='I', LDU≥max1,N;
- otherwise LDU≥1.
- 8: VT(LDVT,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VT
must be at least
max1,N if
COMPQ='I', and at least
1 otherwise.
On exit: if
COMPQ='I', then if
INFO=0, the rows of
VT contain the right singular vectors of the bidiagonal matrix
B.
If
COMPQ≠'I',
VT is not referenced.
- 9: LDVT – INTEGERInput
-
On entry: the first dimension of the array
VT as declared in the (sub)program from which F08MDF (DBDSDC) is called.
Constraints:
- if COMPQ='I', LDVT≥max1,N;
- otherwise LDVT≥1.
- 10: Q(*) – REAL (KIND=nag_wp) arrayOutput
Note: the dimension of the array
Q
must be at least
max1,LDQ, where
LDQ is defined below.
On exit: if
COMPQ='P', then if
INFO=0,
Q and
IQ contain the left and right singular vectors in a compact form, requiring
O
N
logN
space instead of
2×N2. In particular,
Q contains all the real data in the first
LDQ=N×
11+2×smlsiz+8×
int
log2
N/
smlsiz+1
elements of
Q, where
smlsiz is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about
25).
If
COMPQ≠'P',
Q is not referenced.
- 11: IQ(*) – INTEGER arrayOutput
Note: the dimension of the array
IQ
must be at least
max1,LDIQ, where
LDIQ is defined below.
On exit: if
COMPQ='P', then if
INFO=0,
Q and
IQ contain the left and right singular vectors in a compact form, requiring
ONlogN
space instead of
2×N2. In particular,
IQ contains all integer data in the first
LDIQ
=N×
3+3×
int
log2
N/
smlsiz+1
elements of
IQ, where
smlsiz is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about
25).
If
COMPQ≠'P',
IQ is not referenced.
- 12: WORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
WORK
must be at least
max1,LWORK.
Constraints:
- if COMPQ='N', LWORK≥max1,4×N;
- if COMPQ='P', LWORK≥max1,6×N;
- if COMPQ='I', LWORK≥max1,3×N2+4×N.
- 13: IWORK(8×N) – INTEGER arrayWorkspace
- 14: 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 compute a singular value. The update process of divide-and-conquer failed.
7 Accuracy
Each computed singular value of
B is accurate to nearly full relative precision, no matter how tiny the singular value. The
ith computed singular value,
s^i, satisfies the bound
where
ε is the
machine precision and
pn is a modest function of
n.
For bounds on the computed singular values, see Section 4.9.1 of
Anderson et al. (1999). See also
F08FLF (DDISNA).
8 Further Comments
If only singular values are required, the total number of floating point operations is approximately proportional to
n2. When singular vectors are required the number of operations is bounded above by approximately the same number of operations as
F08MEF (DBDSQR), but for large matrices F08MDF (DBDSDC) is usually much faster.
There is no complex analogue of F08MDF (DBDSDC).
9 Example
This example computes the singular value decomposition of the upper bidiagonal matrix
9.1 Program Text
Program Text (f08mdfe.f90)
9.2 Program Data
Program Data (f08mdfe.d)
9.3 Program Results
Program Results (f08mdfe.r)