NAG Library Routine Document

F08KHF (DGEJSV)

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     JOBA – CHARACTER(1)Input

On entry: specifies the form of pivoting for the QR factorization stage; whether an estimate of the condition number of the scaled matrix is required; and the form of rank reduction that is performed.

JOBA='C'

The initial QR factorization of the input matrix is performed with column pivoting; no estimate of condition number is computed; and, the rank is reduced by only the underflowed part of the triangular factor R. This option works well (high relative accuracy) if A=BD, with well-conditioned B and arbitrary diagonal matrix D. The accuracy cannot be spoiled by column scaling. The accuracy of the computed output depends on the condition of B, and the procedure aims at the best theoretical accuracy.

JOBA='E'

Computation as with JOBA='C' with an additional estimate of the condition number of B. It provides a realistic error bound.

JOBA='F'

The initial QR factorization of the input matrix is performed with full row and column pivoting; no estimate of condition number is computed; and, the rank is reduced by only the underflowed part of the triangular factor R. If A=D1×C×D2 with ill-conditioned diagonal scalings D1, D2, and well-conditioned matrix C, this option gives higher accuracy than the JOBA='C' option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable.

JOBA='G'

Computation as with JOBA='F' with an additional estimate of the condition number of B, where A=DB (i.e., B=C×D2). If A has heavily weighted rows, then using this condition number gives too pessimistic an error bound.

JOBA='A'

Computation as with JOBA='C' except in the treatment of rank reduction. In this case, small singular values are to be considered as noise and, if found, the matrix is treated as numerically rank deficient. The computed SVD A=UΣVT restores A up to fm,n×ε×A, where ε is machine precision. This gives the procedure licence to discard (set to zero) all singular values below N×ε×A.

JOBA='R'

Similar to JOBA='A'. The rank revealing property of the initial QR factorization is used to reveal (using the upper triangular factor) a gap σr+1<εσr in which case the numerical rank is declared to be r. The SVD is computed with absolute error bounds, but more accurately than with JOBA='A'.

Constraint: JOBA='C', 'E', 'F', 'G', 'A' or 'R'.

2:     JOBU – CHARACTER(1)Input

On entry: specifies options for computing the left singular vectors U.

JOBU='U'

The first n left singular vectors (columns of U) are computed and returned in the array U.

JOBU='F'

All m left singular vectors are computed and returned in the array U.

JOBU='W'

No left singular vectors are computed, but the array U (with LDU≥M and second dimension at least N) is available as workspace for computing right singular values. See the description of U.

JOBU='N'

No left singular vectors are computed. V is not referenced.

Constraint: JOBU='U', 'F', 'W' or 'N'.

3:     JOBV – CHARACTER(1)Input

On entry: specifies options for computing the right singular vectors V.

JOBV='V'

the n right singular vectors (columns of V) are computed and returned in the array V; Jacobi rotations are not explicitly accumulated.

JOBV='J'

the n right singular vectors (columns of V) are computed and returned in the array V, but they are computed as the product of Jacobi rotations. This option is allowed only if JOBU='U' or 'F', i.e., in computing the full SVD.

JOBV='W'

No right singular values are computed, but the array V (with LDV≥N and second dimension at least N) is available as workspace for computing left singular values. See the description of V.

JOBV='N'

No right singular vectors are computed. U is not referenced.

Constraints:

• JOBV='V', 'J', 'W' or 'N';
• if JOBU='W' or 'N', JOBV≠'J'.

4:     JOBR – CHARACTER(1)Input

On entry: specifies the conditions under which columns of A are to be set to zero. This effectively specifies a lower limit on the range of singular values; any singular values below this limit are (through column zeroing) set to zero. If A≠0 is scaled so that the largest column (in the Euclidean norm) of cA is equal to the square root of the overflow threshold, then JOBR allows the routine to kill columns of A whose norm in cA is less than sfmin (for JOBR='R'), or less than sfmin/ε (otherwise). sfmin is the safe range parameter as returned by routine X02AMF.

JOBR='N'

Only set to zero those columns of A for which the norm of corresponding column of cA<sfmin/ε, that is, those columns that are effectively zero (to machine precision) anyway. If the condition number of A is greater than the overflow threshold λ, where λ is the value returned by X02ALF, you are recommended to use routine F08KJF (DGESVJ).

JOBR='R'

Set to zero those columns of A for which the norm of the corresponding column of cA<sfmin. This approximately represents a restricted range for σcA of sfmin,λ.

For computing the singular values in the full range from the safe minimum up to the overflow threshold use F08KJF (DGESVJ).

Constraint: JOBR='N' or 'R'.

Suggested value: JOBR='R' 

5:     JOBT – CHARACTER(1)Input

On entry: specifies, in the case n=m, whether the routine is permitted to use the transpose of A for improved efficiency. If the matrix is square then the procedure may use transposed A if AT seems to be better with respect to convergence. If the matrix is not square, JOBT is ignored. The decision is based on two values of entropy over the adjoint orbit of ATA. See the descriptions of WORK6 and WORK7.

JOBT='T'

If n=m, perform an entropy test and then transpose if the test indicates possibly faster convergence of the Jacobi process if AT is taken as input. If A is replaced with AT, then the row pivoting is included automatically.

JOBT='N'

No entropy test and no transposition is performed.

The option JOBT='T' can be used to compute only the singular values, or the full SVD (U, Σ and V). In the case where only one set of singular vectors (U or V) is required, the caller must still provide both U and V, as one of the matrices is used as workspace if the matrix A is transposed. See the descriptions of U and V.

Constraint: JOBT='T' or 'N'.

6:     JOBP – CHARACTER(1)Input

On entry: specifies whether the routine should be allowed to introduce structured perturbations to drown denormalized numbers. For details see Drmac and Veselic (2008a) and Drmac and Veselic (2008b). For the sake of simplicity, these perturbations are included only when the full SVD or only the singular values are requested.

JOBP='P'

Introduce perturbation if A is found to be very badly scaled (introducing denormalized numbers).

JOBP='N'

Do not perturb.

Constraint: JOBP='P' or 'N'.

7:     M – INTEGERInput

On entry: m, the number of rows of the matrix A.

Constraint: M≥0.

8:     N – INTEGERInput

On entry: n, the number of columns of the matrix A.

Constraint: M≥N≥0.

9:     A(LDA,*) – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array A must be at least max1,N.

On entry: the m by n matrix A.

On exit: the contents of A are overwritten.

10:   LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F08KHF (DGEJSV) is called.

Constraint: LDA≥max1,M.

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

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

The singular values of A are σi=αSVAi, for i=1,2,…,n, where α=WORK1/WORK2. Normally α=1 and no scaling is required to obtain the singular values. However, if the largest singular value of A overflows or if small singular values have been saved from underflow by scaling the input matrix A, then α≠1.

If JOBR='R' then some of the singular values may be returned as exact zeros because they are below the numerical rank threshold or are denormalized numbers.

12:   U(LDU,*) – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array U must be at least max1,M if JOBU='F', max1,N if JOBU='U' or 'W', and at least 1 otherwise.

On exit: if JOBU='U', U contains the m by n matrix of the left singular vectors.

If JOBU='F', U contains the m by m matrix of the left singular vectors, including an orthonormal basis of the orthogonal complement of Range(A).

If JOBU='W' and (JOBV='V' and JOBT='T' and M=N), then U is used as workspace if the procedure replaces A with AT. In that case, V is computed in U as left singular vectors of AT and then copied back to the array V.

If JOBU='N', U is not referenced.

13:   LDU – INTEGERInput

On entry: the first dimension of the array U as declared in the (sub)program from which F08KHF (DGEJSV) is called.

Constraints:

• if JOBU='U', 'F' or 'W', LDU≥max1,M;
• otherwise LDU≥1.

14:   V(LDV,*) – REAL (KIND=nag_wp) arrayOutput

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

On exit: if JOBV='V' or 'J', V contains the n by n matrix of the right singular vectors.

If JOBV='W' and (JOBU='U' and JOBT='T' and M=N), then V is used as workspace if the procedure replaces A with AT. In that case, U is computed in V as right singular vectors of AT and then copied back to the array U.

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

15:   LDV – INTEGERInput

On entry: the first dimension of the array V as declared in the (sub)program from which F08KHF (DGEJSV) is called.

Constraints:

• if JOBV='V', 'J' or 'W', LDV≥max1,N;
• otherwise LDV≥1.

16:   WORK(LWORK) – REAL (KIND=nag_wp) arrayOutput

On exit: contains information about the completed job.

WORK1

α=WORK1/WORK2 is the scaling factor such that α×SVA1:N are the computed singular values of A. (See the description of SVA.)

WORK2

See the description of WORK1.

WORK3

sconda, an estimate for the condition number of column equilibrated A (if JOBA='E' or 'G'). sconda is an estimate of RTR-11. It is computed using F07FGF (DPOCON). It satisfies n-14×sconda≤R-12≤n14×sconda where R is the triangular factor from the QR factorization of A. However, if R is truncated and the numerical rank is determined to be strictly smaller than n, sconda is returned as -1, thus indicating that the smallest singular values might be lost.

If full SVD is needed, and you are familiar with the details of the method, the following two condition numbers are useful for the analysis of the algorithm.

WORK4

An estimate of the scaled condition number of the triangular factor in the first QR factorization.

WORK5

An estimate of the scaled condition number of the triangular factor in the second QR factorization.

The following two parameters are computed if JOBT='T'.

WORK6

The entropy of ATA: this is the Shannon entropy of diag⁡ATA/trace⁡ATA taken as point in the probability simplex.

WORK7

The entropy of AAT.

17:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F08KHF (DGEJSV) is called.

If JOBU='N' and JOBV='N'

if JOBA≠'E' or 'G'

The minimal requirement is LWORK≥max2M+N,4N+1,7.

For optimal performance the requirement is LWORK≥max2M+N,3N+N+1nb,7, where nb is the block size used by F08AEF (DGEQRF) and F08BFF (DGEQP3). Assuming a value of nb=128 is wise, but choosing a smaller value (e.g., nb=64) should still lead to acceptable performance.

if JOBA='E' or 'G'

In this case, LWORK is the maximum of the above and N×N+4N, i.e., LWORK≥max2M+N,3N+N+1nb,N×N+4N,7.

If JOBU≠'U' or 'F' and JOBV='V' or 'J'

The minimal requirement is LWORK≥max2N+M,7.

For optimal performance, LWORK≥max2N+M,2N+N×nb,7, where nb is described above.

If JOBU='U' or 'F' and JOBV≠'V' or 'J'

The minimal requirement is LWORK≥max2N+M,7.

For optimal performance, LWORK≥max2N+M,2N+N×nb,7, where nb is described above.

If JOBU='U' or 'F' and JOBV='V'

LWORK≥6N+2N×N.

If JOBU='U' or 'F' and JOBV='J'

The minimal requirement is LWORK≥maxM+3N+N×N,7.

For better performance LWORK≥max3N+N×N+M,3N+N×N+N×nb,7, where nb is described above.

18:   IWORK(M+3×N) – INTEGER arrayOutput

On exit: contains information about the completed job.

IWORK1

The numerical rank of A determined after the initial QR factorization with pivoting. See the descriptions of JOBA and JOBR.

IWORK2

The number of computed nonzero singular values.

IWORK3

If nonzero, a warning message: If IWORK3=1 then some of the column norms of A were denormalized (tiny) numbers. The requested high accuracy is not warranted by the data.

19:   INFO – INTEGEROutput

On exit: INFO=0 unless the routine detects an error (see Section 6).

6  Error Indicators and Warnings

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

F08KHF (DGEJSV) did not converge in the maximal number of sweeps. The computed values might be inaccurate.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f08khfe.f90)

9.2  Program Data

Program Data (f08khfe.d)

9.3  Program Results

Program Results (f08khfe.r)