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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgejsv (f08kh)

Purpose

nag_lapack_dgejsv (f08kh) computes the singular value decomposition (SVD) of a real mm by nn matrix AA where mnmn, and optionally computes the left and/or right singular vectors. nag_lapack_dgejsv (f08kh) implements the preconditioned Jacobi SVD of Drmac and Veselic. This is the expert driver function that calls nag_lapack_dgesvj (f08kj) after certain preconditioning. In most cases nag_lapack_dgesvd (f08kb) or nag_lapack_dgesdd (f08kd) is sufficient to obtain the SVD of a real matrix. These are much simpler to use and also handle the case m < nm<n.

Syntax

[a, sva, u, v, work, iwork, info] = f08kh(joba, jobu, jobv, jobr, jobt, jobp, a, 'm', m, 'n', n)
[a, sva, u, v, work, iwork, info] = nag_lapack_dgejsv(joba, jobu, jobv, jobr, jobt, jobp, a, 'm', m, 'n', n)

Description

The SVD is written as
A = UΣVT ,
A = UΣVT ,
where ΣΣ is an mm by nn matrix which is zero except for its nn diagonal elements, UU is an mm by mm orthogonal matrix, and VV is an nn by nn orthogonal matrix. The diagonal elements of ΣΣ are the singular values of AA in descending order of magnitude. The columns of UU and VV are the left and the right singular vectors of AA. The diagonal of ΣΣ is computed and stored in the array sva.

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

Parameters

Compulsory Input Parameters

1:     joba – string (length ≥ 1)
Specifies the form of pivoting for the QRQR 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'joba='C'
The initial QRQR 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 RR. This option works well (high relative accuracy) if A = BDA=BD, with well-conditioned BB and arbitrary diagonal matrix DD. The accuracy cannot be spoiled by column scaling. The accuracy of the computed output depends on the condition of BB, and the procedure aims at the best theoretical accuracy.
joba = 'E'joba='E'
Computation as with joba = 'C'joba='C' with an additional estimate of the condition number of BB. It provides a realistic error bound.
joba = 'F'joba='F'
The initial QRQR 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 RR. If A = D1 × C × D2A=D1×C×D2 with ill-conditioned diagonal scalings D1D1, D2D2, and well-conditioned matrix CC, this option gives higher accuracy than the joba = 'C'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'joba='G'
Computation as with joba = 'F'joba='F' with an additional estimate of the condition number of BB, where A = DBA=DB (i.e., B = C × D2B=C×D2). If AA has heavily weighted rows, then using this condition number gives too pessimistic an error bound.
joba = 'A'joba='A'
Computation as with joba = 'C'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 = UΣVTA=UΣVT restores AA up to f(m,n) × ε × Af(m,n)×ε×A, where εε is machine precision. This gives the procedure licence to discard (set to zero) all singular values below n × ε × An×ε×A.
joba = 'R'joba='R'
Similar to joba = 'A'joba='A'. The rank revealing property of the initial QRQR factorization is used to reveal (using the upper triangular factor) a gap σr + 1 < εσrσr+1<εσr in which case the numerical rank is declared to be rr. The SVD is computed with absolute error bounds, but more accurately than with joba = 'A'joba='A'.
Constraint: joba = 'C'joba='C', 'E''E', 'F''F', 'G''G', 'A''A' or 'R''R'.
2:     jobu – string (length ≥ 1)
Specifies options for computing the left singular vectors UU.
jobu = 'U'jobu='U'
The first nn left singular vectors (columns of UU) are computed and returned in the array u.
jobu = 'F'jobu='F'
All mm left singular vectors are computed and returned in the array u.
jobu = 'W'jobu='W'
No left singular vectors are computed, but the array u (with ldumldum and second dimension at least n) is available as workspace for computing right singular values. See the description of u.
jobu = 'N'jobu='N'
No left singular vectors are computed. uu is not referenced.
Constraint: jobu = 'U'jobu='U', 'F''F', 'W''W' or 'N''N'.
3:     jobv – string (length ≥ 1)
Specifies options for computing the right singular vectors VV.
jobv = 'V'jobv='V'
the nn right singular vectors (columns of VV) are computed and returned in the array v; Jacobi rotations are not explicitly accumulated.
jobv = 'J'jobv='J'
the nn right singular vectors (columns of VV) 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'jobu='U' or 'F''F', i.e., in computing the full SVD.
jobv = 'W'jobv='W'
No right singular values are computed, but the array v (with ldvnldvn and second dimension at least n) is available as workspace for computing left singular values. See the description of v.
jobv = 'N'jobv='N'
No right singular vectors are computed. vv is not referenced.
Constraints:
  • jobv = 'V'jobv='V', 'J''J', 'W''W' or 'N''N';
  • if jobu = 'W'jobu='W' or 'N''N', jobv'J'jobv'J'.
4:     jobr – string (length ≥ 1)
Specifies the conditions under which columns of AA 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 A0A0 is scaled so that the largest column (in the Euclidean norm) of cAcA is equal to the square root of the overflow threshold, then jobr allows the function to kill columns of AA whose norm in cAcA is less than sqrt(sfmin)sfmin (for jobr = 'R'jobr='R'), or less than sfmin / εsfmin/ε (otherwise). sfminsfmin is the safe range parameter, as returned by function nag_machine_real_safe (x02am).
jobr = 'N'jobr='N'
Only set to zero those columns of AA for which the norm of corresponding column of cA < sfmin / εcA<sfmin/ε, that is, those columns that are effectively zero (to machine precision) anyway. If the condition number of AA is greater than the overflow threshold λλ, where λλ is the value returned by nag_machine_real_largest (x02al), you are recommended to use function nag_lapack_dgesvj (f08kj).
jobr = 'R'jobr='R'
Set to zero those columns of AA for which the norm of the corresponding column of cA < sqrt(sfmin)cA<sfmin. This approximately represents a restricted range for σ(cA)σ(cA) of [sqrt(sfmin),sqrt(λ)][sfmin,λ].
For computing the singular values in the full range from the safe minimum up to the overflow threshold use nag_lapack_dgesvj (f08kj)
Constraint: jobr = 'N'jobr='N' or 'R''R'.
5:     jobt – string (length ≥ 1)
Specifies, in the case n = mn=m, whether the function is permitted to use the transpose of AA for improved efficiency. If the matrix is square then the procedure may use transposed AA if ATAT 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 ATAATA. See the descriptions of work(6)work6 and work(7)work7.
jobt = 'T'jobt='T'
If n = mn=m, perform an entropy test and then transpose if the test indicates possibly faster convergence of the Jacobi process if ATAT is taken as input. If AA is replaced with ATAT, then the row pivoting is included automatically.
jobt = 'N'jobt='N'
No entropy test and no transposition is performed.
The option jobt = 'T'jobt='T' can be used to compute only the singular values, or the full SVD (UU, ΣΣ and VV). In the case where only one set of singular vectors (UU or VV) is required, the caller must still provide both u and v, as one of the matrices is used as workspace if the matrix AA is transposed. See the descriptions of u and v
Constraint: jobt = 'T'jobt='T' or 'N''N'.
6:     jobp – string (length ≥ 1)
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 = 'P'jobp='P'
Introduce perturbation if AA is found to be very badly scaled (introducing denormalized numbers).
jobp = 'N'jobp='N'
Do not perturb.
Constraint: jobp = 'P'jobp='P' or 'N''N'.
7:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,m)max(1,m)
The second dimension of the array must be at least max (1,n)max(1,n)
The mm by nn matrix AA.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The first dimension of the array a.
mm, the number of rows of the matrix AA.
Constraint: m0m0.
2:     n – int64int32nag_int scalar
Default: The second dimension of the array a.
nn, the number of columns of the matrix AA.
Constraint: mn0mn0.

Input Parameters Omitted from the MATLAB Interface

lda ldu ldv lwork

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,m)max(1,m)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,m)ldamax(1,m).
The contents of a are overwritten.
2:     sva(n) – double array
The, possibly scaled, singular values of AA.
The singular values of AA are σi = αsva(i)σi=αsvai, for i = 1,2,,ni=1,2,,n, where α = work(1) / work(2)α=work1/work2. Normally α = 1α=1 and no scaling is required to obtain the singular values. However, if the largest singular value of AA overflows or if small singular values have been saved from underflow by scaling the input matrix AA, then α1α1.
If jobr = 'R'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.
3:     u(ldu, : :) – double array
The first dimension, ldu, of the array u will be
  • if jobu = 'F'jobu='F', 'U''U' or 'W''W', ldumax (1,m)ldumax(1,m);
  • otherwise ldu1ldu1.
The second dimension of the array will be max (1,m)max(1,m) if jobu = 'F'jobu='F', max (1,n)max(1,n) if jobu = 'U'jobu='U' or 'W''W', and at least 11 otherwise
If jobu = 'U'jobu='U', u contains the mm by nn matrix of the left singular vectors.
If jobu = 'F'jobu='F', u contains the mm by mm matrix of the left singular vectors, including an orthonormal basis of the orthogonal complement of Range(AA).
If jobu = 'W'jobu='W' and (jobv = 'V'jobv='V' and jobt = 'T'jobt='T' and m = nm=n), then u is used as workspace if the procedure replaces AA with ATAT. In that case, VV is computed in u as left singular vectors of ATAT and then copied back to the array v.
If jobu = 'N'jobu='N', u is not referenced.
4:     v(ldv, : :) – double array
The first dimension, ldv, of the array v will be
  • if jobv = 'V'jobv='V', 'J''J' or 'W''W', ldvmax (1,n)ldvmax(1,n);
  • otherwise ldv1ldv1.
The second dimension of the array will be max (1,n)max(1,n) if jobv = 'V'jobv='V', 'J''J' or 'W''W', and at least 11 otherwise
If jobv = 'V'jobv='V' or 'J''J', v contains the nn by nn matrix of the right singular vectors.
If jobv = 'W'jobv='W' and (jobu = 'U'jobu='U' and jobt = 'T'jobt='T' and m = nm=n), then v is used as workspace if the procedure replaces AA with ATAT. In that case, UU is computed in v as right singular vectors of ATAT and then copied back to the array u.
If jobv = 'N'jobv='N', v is not referenced.
5:     work(lwork) – double array
Contains information about the completed job.
work(1)work1
α = work(1) / work(2)α=work1/work2 is the scaling factor such that σi = αsva(i)σi=αsvai, for i = 1,2,,ni=1,2,,n are the computed singular values of AA. (See the description of svasva.)
work(2)work2
See the description of work(1)work1.
work(3)work3
sconda, an estimate for the condition number of column equilibrated AA (if joba = 'E'joba='E' or 'G''G'). sconda is an estimate of sqrt(((RTR)11))((RTR)-11). It is computed using nag_lapack_dpocon (f07fg). It satisfies n(1/4) × scondaR12n(1/4) × scondan-14×scondaR-12n14×sconda where RR is the triangular factor from the QRQR factorization of AA. However, if RR is truncated and the numerical rank is determined to be strictly smaller than nn, sconda is returned as 1-1, 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(4)work4
An estimate of the scaled condition number of the triangular factor in the first QRQR factorization.
work(5)work5
An estimate of the scaled condition number of the triangular factor in the second QRQR factorization.
The following two parameters are computed if jobt = 'T'jobt='T'.
work(6)work6
The entropy of ATAATA: this is the Shannon entropy of diagATA / traceATAdiagATA/traceATA taken as a point in the probability simplex.
work(7)work7
The entropy of AATAAT.
6:     iwork(m + 3 × nm+3×n) – int64int32nag_int array
Contains information about the completed job.
iwork(1)iwork1
The numerical rank of AA determined after the initial QRQR factorization with pivoting. See the descriptions of joba and jobr.
iwork(2)iwork2
The number of computed nonzero singular values.
iwork(3)iwork3
If nonzero, a warning message: If iwork(3) = 1iwork3=1 then some of the column norms of AA were denormalized (tiny) numbers. The requested high accuracy is not warranted by the data.
7:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W INFO > 0INFO>0
nag_lapack_dgejsv (f08kh) did not converge in the allowed number of iterations (3030). The computed values might be inaccurate.

Accuracy

The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix (A + E) (A+E) , where
E2 = O(ε) A2 ,
E2 = O(ε) A2 ,
and ε 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.

Further Comments

nag_lapack_dgejsv (f08kh) implements a preconditioned Jacobi SVD algorithm. It uses nag_lapack_dgeqrf (f08ae), nag_lapack_dgelqf (f08ah) and nag_lapack_dgeqp3 (f08bf) 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 AA with the structure A = D1CD2A=D1CD2, where D1D1, D2D2 are arbitrarily ill-conditioned diagonal matrices and CC is a well-conditioned matrix. In that case, complete pivoting in the first QRQR factorizations provides accuracy dependent on the condition number of CC, and independent of D1D1, D2D2. Such higher accuracy is not completely understood theoretically, but it works well in practice. Further, if AA can be written as A = BDA=BD, with well-conditioned BB and some diagonal DD, then the high accuracy is guaranteed, both theoretically and in software, independent of DD.

Example

function nag_lapack_dgejsv_example
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];
joba = 'e';
jobu = 'u';
jobv = 'v';
jobr = 'r';
jobt = 'n';
jobp = 'n';
% Compute the singular values and left and right singular vectors
% of A (A = U*S*V^T, m >= n)
[a, sva, u, v, work, iwork, info] = nag_lapack_dgejsv(joba, jobu, jobv, jobr, jobt, jobp, a);

if info == 0
  % Compute the approximate error bound for the computed singular values
  % using the 2-norm, s(1) = norm(A), and machine precision, eps.
  eps = nag_machine_precision;
  serrbd = eps*sva(1);

  % Print solution
  if (abs(work(1)-work(2)) < 2*eps)
    % No scaling required
    fprintf('\nSingular values:\n');
    disp(transpose(sva));
  else
    fprintf('\nScaled singular values:\n');
    disp(transpose(sva));
    fprintf('\nFor true singular values, multiply by a/b\n');
    fprintf('   where a=%13.5e and b=%13.5e.\n', work(1), work(2));
  end

  fprintf('\n');
  [ifail] = nag_file_print_matrix_real_gen('g', ' ', u, 'Left singular vectors');
  fprintf('\n');
  [ifail] = nag_file_print_matrix_real_gen('g', ' ', v, 'Right singular vectors');

  % Call nag_lapack_ddisna to estimate reciprocal condition numbers for the singular vectors
  [rcondu, info] = nag_lapack_ddisna('Left', int64(6), int64(4), sva);
  [rcondv, info] = nag_lapack_ddisna('Right', int64(6), int64(4), sva);

  % Print the approximate error bounds for the singular values and vectors
  fprintf('\nEstimate of the condition number of column equilibrated a\n');
  fprintf('%11.1e\n', work(3));
  fprintf('\nError estimate for the singular values a\n');
  fprintf('%11.1e\n', serrbd);
  fprintf('\nError estimates for left singular vectors\n');
  fprintf('%11.1e ',serrbd./rcondu);
  fprintf('\n\nError estimates for right singular vectors\n');
  fprintf('%11.1e ',serrbd./rcondv);
  fprintf('\n');
end
 

Singular values:
    9.9966    3.6831    1.3569    0.5000


 Left singular vectors
          1       2       3       4
 1   0.2774 -0.6003 -0.1277  0.1323
 2   0.2020 -0.0301  0.2805  0.7034
 3   0.2918  0.3348  0.6453  0.1906
 4  -0.0938 -0.3699  0.6781 -0.5399
 5  -0.4213  0.5266  0.0413 -0.0575
 6   0.7816  0.3353 -0.1645 -0.3957

 Right singular vectors
          1       2       3       4
 1   0.1921 -0.8030  0.0041 -0.5642
 2  -0.8794 -0.3926 -0.0752  0.2587
 3   0.2140 -0.2980  0.7827  0.5027
 4  -0.3795  0.3351  0.6178 -0.6017

Estimate of the condition number of column equilibrated a
    9.0e+00

Error estimate for the singular values a
    1.1e-15

Error estimates for left singular vectors
    1.8e-16     4.8e-16     1.3e-15     2.2e-15 

Error estimates for right singular vectors
    1.8e-16     4.8e-16     1.3e-15     1.3e-15 

function f08kh_example
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];
joba = 'e';
jobu = 'u';
jobv = 'v';
jobr = 'r';
jobt = 'n';
jobp = 'n';
% Compute the singular values and left and right singular vectors
% of A (A = U*S*V^T, m >= n)
[a, sva, u, v, work, iwork, info] = f08kh(joba, jobu, jobv, jobr, jobt, jobp, a);

if info == 0
  % Compute the approximate error bound for the computed singular values
  % using the 2-norm, s(1) = norm(A), and machine precision, eps.
  eps = x02aj;
  serrbd = eps*sva(1);

  % Print solution
  if (abs(work(1)-work(2)) < 2*eps)
    % No scaling required
    fprintf('\nSingular values:\n');
    disp(transpose(sva));
  else
    fprintf('\nScaled singular values:\n');
    disp(transpose(sva));
    fprintf('\nFor true singular values, multiply by a/b\n');
    fprintf('   where a=%13.5e and b=%13.5e.\n', work(1), work(2));
  end

  fprintf('\n');
  [ifail] = x04ca('g', ' ', u, 'Left singular vectors');
  fprintf('\n');
  [ifail] = x04ca('g', ' ', v, 'Right singular vectors');

  % Call f08fl to estimate reciprocal condition numbers for the singular vectors
  [rcondu, info] = f08fl('Left', int64(6), int64(4), sva);
  [rcondv, info] = f08fl('Right', int64(6), int64(4), sva);

  % Print the approximate error bounds for the singular values and vectors
  fprintf('\nEstimate of the condition number of column equilibrated a\n');
  fprintf('%11.1e\n', work(3));
  fprintf('\nError estimate for the singular values a\n');
  fprintf('%11.1e\n', serrbd);
  fprintf('\nError estimates for left singular vectors\n');
  fprintf('%11.1e ',serrbd./rcondu);
  fprintf('\n\nError estimates for right singular vectors\n');
  fprintf('%11.1e ',serrbd./rcondv);
  fprintf('\n');
end
 

Singular values:
    9.9966    3.6831    1.3569    0.5000


 Left singular vectors
          1       2       3       4
 1   0.2774 -0.6003 -0.1277  0.1323
 2   0.2020 -0.0301  0.2805  0.7034
 3   0.2918  0.3348  0.6453  0.1906
 4  -0.0938 -0.3699  0.6781 -0.5399
 5  -0.4213  0.5266  0.0413 -0.0575
 6   0.7816  0.3353 -0.1645 -0.3957

 Right singular vectors
          1       2       3       4
 1   0.1921 -0.8030  0.0041 -0.5642
 2  -0.8794 -0.3926 -0.0752  0.2587
 3   0.2140 -0.2980  0.7827  0.5027
 4  -0.3795  0.3351  0.6178 -0.6017

Estimate of the condition number of column equilibrated a
    9.0e+00

Error estimate for the singular values a
    1.1e-15

Error estimates for left singular vectors
    1.8e-16     4.8e-16     1.3e-15     2.2e-15 

Error estimates for right singular vectors
    1.8e-16     4.8e-16     1.3e-15     1.3e-15 


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