nag_dgejsv (f08khc) (PDF version)
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f08 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_dgejsv (f08khc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_dgejsv (f08khc) computes the singular value decomposition (SVD) of a real m by n matrix A where mn, and optionally computes the left and/or right singular vectors. nag_dgejsv (f08khc) implements the preconditioned Jacobi SVD of Drmac and Veselic. This is the expert driver function that calls nag_dgesvj (f08kjc) after certain preconditioning. In most cases nag_dgesvd (f08kbc) or nag_dgesdd (f08kdc) is sufficient to obtain the SVD of a real matrix. These are much simpler to use and also handle the case m<n.

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_dgejsv (Nag_OrderType order, Nag_Preprocess joba, Nag_LeftVecsType jobu, Nag_RightVecsType jobv, Nag_ZeroCols jobr, Nag_TransType jobt, Nag_Perturb jobp, Integer m, Integer n, double a[], Integer pda, double sva[], double u[], Integer pdu, double v[], Integer pdv, double work[], Integer iwork[], NagError *fail)

3  Description

The SVD is written as
A = UΣVT ,
where Σ is an m by n matrix which is zero except for its n diagonal elements, U is an m by m orthogonal matrix, and V is an n by n orthogonal matrix. The diagonal elements of Σ are the singular values of A in descending order of magnitude. The columns of U and V are the left and the right singular vectors of A. The diagonal of Σ is computed and stored in the array sva.

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
Drmac Z and Veselic K (2008a) New fast and accurate Jacobi SVD algorithm I SIAM J. Matrix Anal. Appl. 29 4
Drmac Z and Veselic K (2008b) New fast and accurate Jacobi SVD algorithm II SIAM J. Matrix Anal. Appl. 29 4
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     jobaNag_PreprocessInput
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=Nag_ColpivRrank
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=Nag_ColpivRrankCond
Computation as with joba=Nag_ColpivRrank with an additional estimate of the condition number of B. It provides a realistic error bound.
joba=Nag_FullpivRrank
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=Nag_ColpivRrank option. If the structure of the input matrix is not known, and relative accuracy is desirable, then this option is advisable.
joba=Nag_FullpivRrankCond
Computation as with joba=Nag_FullpivRrank 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=Nag_ColpivSVrankAbs
Computation as with joba=Nag_ColpivRrank 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=Nag_ColpivSVrankRel
Similar to joba=Nag_ColpivSVrankAbs. 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=Nag_ColpivSVrankAbs.
Constraint: joba=Nag_ColpivRrank, Nag_ColpivRrankCond, Nag_FullpivRrank, Nag_FullpivRrankCond, Nag_ColpivSVrankAbs or Nag_ColpivSVrankRel.
3:     jobuNag_LeftVecsTypeInput
On entry: specifies options for computing the left singular vectors U.
jobu=Nag_LeftSpan
The first n left singular vectors (columns of U) are computed and returned in the array u.
jobu=Nag_LeftVecs
All m left singular vectors are computed and returned in the array u.
jobu=Nag_NotLeftWork
No left singular vectors are computed, but the array u (with pdum and second dimension at least n) is available as workspace for computing right singular values. See the description of u.
jobu=Nag_NotLeftVecs
No left singular vectors are computed. u is not referenced.
Constraint: jobu=Nag_LeftSpan, Nag_LeftVecs, Nag_NotLeftWork or Nag_NotLeftVecs.
4:     jobvNag_RightVecsTypeInput
On entry: specifies options for computing the right singular vectors V.
jobv=Nag_RightVecs
the n right singular vectors (columns of V) are computed and returned in the array v; Jacobi rotations are not explicitly accumulated.
jobv= Nag_RightVecsJRots
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=Nag_LeftSpan or Nag_LeftVecs, i.e., in computing the full SVD.
jobv=Nag_NotRightWork
No right singular values are computed, but the array v (with pdvn and second dimension at least n) is available as workspace for computing left singular values. See the description of v.
jobv=Nag_NotRightVecs
No right singular vectors are computed. v is not referenced.
Constraints:
  • jobv=Nag_RightVecs, Nag_RightVecsJRots, Nag_NotRightWork or Nag_NotRightVecs;
  • if jobu=Nag_NotLeftWork or Nag_NotLeftVecs, jobv Nag_RightVecsJRots.
5:     jobrNag_ZeroColsInput
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 A0 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 function to kill columns of A whose norm in cA is less than sfmin (for jobr=Nag_ZeroColsRestrict), or less than sfmin/ε (otherwise). sfmin is the safe range argument as returned by function nag_real_safe_small_number (X02AMC).
jobr=Nag_ZeroColsNormal
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 nag_real_largest_number (X02ALC), you are recommended to use function nag_dgesvj (f08kjc).
jobr=Nag_ZeroColsRestrict
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 nag_dgesvj (f08kjc).
Suggested value: jobr=Nag_ZeroColsRestrict 
Constraint: jobr=Nag_ZeroColsNormal or Nag_ZeroColsRestrict.
6:     jobtNag_TransTypeInput
On entry: specifies, in the case n=m, whether the function 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 work[5] and work[6].
jobt=Nag_Trans
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=Nag_NoTrans
No entropy test and no transposition is performed.
The option jobt=Nag_Trans 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=Nag_Trans or Nag_NoTrans.
7:     jobpNag_PerturbInput
On entry: specifies whether the function 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=Nag_PerturbOn
Introduce perturbation if A is found to be very badly scaled (introducing denormalized numbers).
jobp=Nag_PerturbOff
Do not perturb.
Constraint: jobp=Nag_PerturbOn or Nag_PerturbOff.
8:     mIntegerInput
On entry: m, the number of rows of the matrix A.
Constraint: m0.
9:     nIntegerInput
On entry: n, the number of columns of the matrix A.
Constraint: mn0.
10:   a[dim]doubleInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×n when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix A.
On exit: the contents of a are overwritten.
11:   pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,m;
  • if order=Nag_RowMajor, pdamax1,n.
12:   sva[n]doubleOutput
On exit: the, possibly scaled, singular values of A.
The singular values of A are σi=αsva[i-1], for i=1,2,,n, where α=work[0]/work[1]. 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 α1.
If jobr=Nag_ZeroColsRestrict then some of the singular values may be returned as exact zeros because they are below the numerical rank threshold or are denormalized numbers.
13:   u[dim]doubleOutput
Note: the dimension, dim, of the array u must be at least
  • max1,pdu×m when jobu=Nag_LeftVecs;
  • max1,pdu×n when jobu=Nag_LeftSpan or Nag_NotLeftWork and order=Nag_ColMajor;
  • max1,m×pdu when jobu=Nag_LeftSpan or Nag_NotLeftWork and order=Nag_RowMajor;
  • 1 otherwise.
The i,jth element of the matrix U is stored in
  • u[j-1×pdu+i-1] when order=Nag_ColMajor;
  • u[i-1×pdu+j-1] when order=Nag_RowMajor.
On exit: if jobu=Nag_LeftSpan, u contains the m by n matrix of the left singular vectors.
If jobu=Nag_LeftVecs, 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=Nag_NotLeftWork and (jobv=Nag_RightVecs and jobt=Nag_Trans 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=Nag_NotLeftVecs, u is not referenced.
14:   pduIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array u.
Constraints:
  • if order=Nag_ColMajor,
    • if jobu=Nag_LeftVecs, pdumax1,m;
    • if jobu=Nag_LeftSpan or Nag_NotLeftWork, pdumax1,m;
    • otherwise pdu1;
  • if order=Nag_RowMajor,
    • if jobu=Nag_LeftVecs, pdumax1,m;
    • if jobu=Nag_LeftSpan or Nag_NotLeftWork, pdumax1,n;
    • otherwise pdu1.
15:   v[dim]doubleOutput
Note: the dimension, dim, of the array v must be at least
  • max1,pdv×n when jobv=Nag_RightVecs, Nag_RightVecsJRots or Nag_NotRightWork;
  • 1 otherwise.
The i,jth element of the matrix V is stored in
  • v[j-1×pdv+i-1] when order=Nag_ColMajor;
  • v[i-1×pdv+j-1] when order=Nag_RowMajor.
On exit: if jobv=Nag_RightVecs or Nag_RightVecsJRots, v contains the n by n matrix of the right singular vectors.
If jobv=Nag_NotRightWork and (jobu=Nag_LeftSpan and jobt=Nag_Trans 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=Nag_NotRightVecs, v is not referenced.
16:   pdvIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array v.
Constraints:
  • if jobv=Nag_RightVecs, Nag_RightVecsJRots or Nag_NotRightWork, pdvmax1,n;
  • otherwise pdv1.
17:   work[7]doubleOutput
On exit: contains information about the completed job.
work[0]
α=work[0]/work[1] is the scaling factor such that σi=αsva[i-1], for i=1,2,,n are the computed singular values of A. (See the description of sva.)
work[1]
See the description of work[0].
work[2]
sconda, an estimate for the condition number of column equilibrated A (if joba=Nag_ColpivRrankCond or Nag_FullpivRrankCond). sconda is an estimate of RTR-11. It is computed using nag_dpocon (f07fgc). It satisfies n-14×scondaR-12n14×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.
work[3]
An estimate of the scaled condition number of the triangular factor in the first QR factorization.
work[4]
An estimate of the scaled condition number of the triangular factor in the second QR factorization.
The following two parameters are computed if jobt=Nag_Trans.
work[5]
The entropy of ATA: this is the Shannon entropy of diagATA/traceATA taken as a point in the probability simplex.
work[6]
The entropy of AAT.
18:   iwork[3]IntegerOutput
On exit: contains information about the completed job.
iwork[0]
The numerical rank of A determined after the initial QR factorization with pivoting. See the descriptions of joba and jobr.
iwork[1]
The number of computed nonzero singular values.
iwork[2]
If nonzero, a warning message: If iwork[2]=1 then some of the column norms of A were denormalized (tiny) numbers. The requested high accuracy is not warranted by the data.
19:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_CONVERGENCE
nag_dgejsv (f08khc) did not converge in the allowed number of iterations (30). The computed values might be inaccurate.
NE_ENUM_INT_2
On entry, jobu=value, m=value and pdu=value.
Constraint: if jobu=Nag_LeftVecs, pdumax1,m;
if jobu=Nag_LeftSpan or Nag_NotLeftWork, pdumax1,m;
otherwise pdu1.
On entry, jobv=value, pdv=value, n=value.
Constraint: if jobv=Nag_RightVecs, Nag_RightVecsJRots or Nag_NotRightWork, pdvmax1,n;
otherwise pdv1.
On entry, pdv=value, jobv=value and n=value.
Constraint: if jobv=Nag_RightVecs, Nag_RightVecsJRots or Nag_NotRightWork, pdvmax1,n;
otherwise pdv1.
NE_ENUM_INT_3
On entry, jobu=value, pdu=value, m=value and n=value.
Constraint: if jobu=Nag_LeftVecs, pdumax1,m;
if jobu=Nag_LeftSpan or Nag_NotLeftWork, pdumax1,n;
otherwise pdu1.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdu=value.
Constraint: pdu>0.
On entry, pdv=value.
Constraint: pdv>0.
NE_INT_2
On entry, m=value and n=value.
Constraint: mn0.
On entry, pda=value and m=value.
Constraint: pdamax1,m.
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

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.

8  Further Comments

nag_dgejsv (f08khc) implements a preconditioned Jacobi SVD algorithm. It uses nag_dgeqrf (f08aec), nag_dgelqf (f08ahc) and nag_dgeqp3 (f08bfc) as preprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor, which in some cases results in much higher accuracy. An example is matrix A with the structure A=D1CD2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is a well-conditioned matrix. In that case, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number of C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if A can be written as A=BD, with well-conditioned B and some diagonal D, then the high accuracy is guaranteed, both theoretically and in software, independent of D.

9  Example

This example finds the singular values and left and right singular vectors of the 6 by 4 matrix
A = 2.27 -1.54 1.15 -1.94 0.28 -1.67 0.94 -0.78 -0.48 -3.09 0.99 -0.21 1.07 1.22 0.79 0.63 -2.35 2.93 -1.45 2.30 0.62 -7.39 1.03 -2.57 ,
together with the condition number of A and approximate error bounds for the computed singular values and vectors.

9.1  Program Text

Program Text (f08khce.c)

9.2  Program Data

Program Data (f08khce.d)

9.3  Program Results

Program Results (f08khce.r)


nag_dgejsv (f08khc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012