F08MDF (DBDSDC) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

F08MDF (DBDSDC)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

+ Contents

    1  Purpose
    7  Accuracy

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
B = USVT ,
where S is a diagonal matrix with non-negative diagonal elements sii=si, such that
s1 s2 sn 0 ,
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/lug
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 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: N0.
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', LDUmax1,N;
  • otherwise LDU1.
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', LDVTmax1,N;
  • otherwise LDVT1.
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', LWORKmax1,4×N;
  • if COMPQ='P', LWORKmax1,6×N;
  • if COMPQ='I', LWORKmax1,3×N2+4×N.
13:   IWORK(8×N) – INTEGER arrayWorkspace
14:   INFO – INTEGEROutput

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
s^i-si pnεsi
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
B = 3.62 1.26 0.00 0.00 0.00 -2.41 -1.53 0.00 0.00 0.00 1.92 1.19 0.00 0.00 0.00 -1.43 .

9.1  Program Text

Program Text (f08mdfe.f90)

9.2  Program Data

Program Data (f08mdfe.d)

9.3  Program Results

Program Results (f08mdfe.r)


F08MDF (DBDSDC) (PDF version)
F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2011