NAG Library Routine Document

F08KJF (DGESVJ)

if INFO=0: rankA orthonormal columns of U are returned in the leading rankA columns of the array A. Here rankA≤N is the number of computed singular values of A that are above the safe range parameter as returned by X02AMF. The singular vectors corresponding to underflowed or zero singular values are not computed. The value of rankA is returned by rounding WORK2 to the nearest whole number. Also see the descriptions of SVA and WORK. The computed columns of U are mutually numerically orthogonal up to approximately tol=m×ε; or tol=ctol×ε (JOBU='C'), where ε is the machine precision and ctol is supplied on entry in WORK1, see the description of JOBU.
If INFO>0: F08KJF (DGESVJ) did not converge in 30 iterations (sweeps). In this case, the computed columns of U may not be orthogonal up to tol. The output U (stored in A), Σ (given by the computed singular values in SVA) and V is still a decomposition of the input matrix A in the sense that the residual A-α×U×Σ×VT2/A2 is small, where α is the value returned in WORK1.

If JOBU='N'

if INFO=0: Note that the left singular vectors are ‘for free’ in the one-sided Jacobi SVD algorithm. However, if only the singular values are needed, the level of numerical orthogonality of U is not an issue and iterations are stopped when the columns of the iterated matrix are numerically orthogonal up to approximately m×ε. Thus, on exit, A contains the columns of U scaled with the corresponding singular values.
If INFO>0: F08KJF (DGESVJ) did not converge in 30 iterations (sweeps).

7: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08KJF (DGESVJ) is called.

Constraint: LDA≥max1,M.

8: SVA(N) – REAL (KIND=nag_wp) arrayOutput

On exit: the, possibly scaled, singular values of A.

If INFO=0: The singular values of A are σi=αSVAi, for i=1,2,…,n, where α is the scale factor stored in WORK1. Normally α=1, however, if some of the singular values of A might underflow or overflow, then α≠1 and the scale factor needs to be applied to obtain the singular values.
If INFO>0: F08KJF (DGESVJ) did not converge in 30 iterations and α×SVA may not be accurate.

9: MV – INTEGERInput

On entry: if JOBV='A', the product of Jacobi rotations is applied to the first MV rows of V.

If JOBV≠'A', MV is ignored. See the description of JOBV.

10: V(LDV,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array V must be at least max1,N if JOBV='V' or 'A', and at least 1 otherwise.

On entry: if JOBV='A', V must contain an MV by n matrix to be premultiplied by the matrix V of right singular vectors.

On exit: the right singular vectors of A.

If JOBV='V', V contains the n by n matrix of the right singular vectors.

If JOBV='A', V contains the product of the computed right singular vector matrix and the initial matrix in the array V.

If JOBV='N', V is not referenced.

11: LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F08KJF (DGESVJ) is called.

Constraints:

if JOBV='V', LDV≥max1,N;
if JOBV='A', LDV≥max1,MV;
otherwise LDV≥1.

12: WORK(LWORK) – REAL (KIND=nag_wp) arrayCommunication Array

On entry: if JOBU='C', WORK1=ctol, where ctol defines the threshold for convergence. The process stops if all columns of A are mutually orthogonal up to ctol×ε. It is required that ctol≥1, i.e., it is not possible to force the routine to obtain orthogonality below ε. ctol greater than 1/ε is meaningless, where ε is the machine precision.

On exit: contains information about the completed job.

WORK1: the scaling factor, α, such that α×SVA1:N are the computed singular values of A. (See description of SVA.)
WORK2: nintWORK2gives the number of the computed nonzero singular values.
WORK3: nintWORK3 gives the number of the computed singular values that are larger than the underflow threshold.
WORK4: nintWORK4 gives the number of iterations (sweeps of Jacobi rotations) needed for numerical convergence.
WORK5: maxi≠jcosA:,i,A:,j in the last iteration (sweep). This is useful information in cases when F08KJF (DGESVJ) did not converge, as it can be used to estimate whether the output is still useful and for subsequent analysis.
WORK6: The largest absolute value over all sines of the Jacobi rotation angles in the last sweep. It can be useful for subsequent analysis.

Constraint: if JOBU='C', WORK1≥1.0.

13: LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08KJF (DGESVJ) is called.

Constraint: LWORK≥max6,M+N.

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: F08KJF (DGESVJ) did not converge in 30 iterations (sweeps), but its output might still be useful.

7 Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix A+E , where

E2 = Oε A2 ,

and ε is the machine precision. In addition, the computed singular vectors are nearly orthogonal to working precision. See Section 4.9 of Anderson et al. (1999) for further details.

See Section 6 of Drmac and Veselic (2008a) for a detailed discussion of the accuracy of the computed SVD.

8 Further Comments

This SVD algorithm is numerically superior to the bidiagonalization based QR algorithm implemented by F08KBF (DGESVD) and the divide and conquer algorithm implemented by F08KDF (DGESDD) algorithms and is considerably faster than previous implementations of the (equally accurate) Jacobi SVD method. Moreover, this algorithm can compute the SVD faster than F08KBF (DGESVD) and not much slower than F08KDF (DGESDD). See Section 3.3 of Drmac and Veselic (2008b) for the details.