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NAG Toolbox: nag_eigen_real_triang_svd (f02wu)

Purpose

nag_eigen_real_triang_svd (f02wu) returns all, or part, of the singular value decomposition of a real upper triangular matrix.

Syntax

[a, b, q, sv, work, ifail] = f02wu(a, b, wantq, wantp, 'n', n, 'ncolb', ncolb)
[a, b, q, sv, work, ifail] = nag_eigen_real_triang_svd(a, b, wantq, wantp, 'n', n, 'ncolb', ncolb)

Description

The nn by nn upper triangular matrix RR is factorized as
R = QSPT,
R=QSPT,
where QQ and PP are nn by nn orthogonal matrices and SS is an nn by nn diagonal matrix with non-negative diagonal elements, σ1,σ2,,σnσ1,σ2,,σn, ordered such that
σ1σ2σn0.
σ1σ2σn0.
The columns of QQ are the left-hand singular vectors of RR, the diagonal elements of SS are the singular values of RR and the columns of PP are the right-hand singular vectors of RR.
Either or both of QQ and PTPT may be requested and the matrix CC given by
C = QTB,
C=QTB,
where BB is an nn by ncolbncolb given matrix, may also be requested.
The function obtains the singular value decomposition by first reducing RR to bidiagonal form by means of Givens plane rotations and then using the QRQR algorithm to obtain the singular value decomposition of the bidiagonal form.
Good background descriptions to the singular value decomposition are given in Chan (1982), Dongarra et al. (1979), Golub and Van Loan (1996), Hammarling (1985) and Wilkinson (1978).
Note that if KK is any orthogonal diagonal matrix so that
KKT = I
KKT=I
(that is the diagonal elements of KK are + 1+1 or 1-1) then
A = (QK)S(PK)T
A=(QK)S(PK)T
is also a singular value decomposition of AA.

References

Chan T F (1982) An improved algorithm for computing the singular value decomposition ACM Trans. Math. Software 8 72–83
Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25
Wilkinson J H (1978) Singular Value Decomposition – Basic Aspects Numerical Software – Needs and Availability (ed D A H Jacobs) Academic Press

Parameters

Compulsory Input Parameters

1:     a(lda, : :) – double array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The leading nn by nn upper triangular part of the array a must contain the upper triangular matrix RR.
2:     b(ldb, : :) – double array
The first dimension, ldb, of the array b must satisfy
  • if ncolb > 0ncolb>0, ldbmax (1,n)ldbmax(1,n);
  • otherwise ldb1ldb1.
The second dimension of the array must be at least max (1,ncolb)max(1,ncolb)
With ncolb > 0ncolb>0, the leading nn by ncolbncolb part of the array b must contain the matrix to be transformed.
3:     wantq – logical scalar
Must be true if the matrix QQ is required.
If wantq = falsewantq=false, the array q is not referenced.
4:     wantp – logical scalar
Must be true if the matrix PTPT is required, in which case PTPT is overwritten on the array a, otherwise wantp must be false.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
nn, the order of the matrix RR.
If n = 0n=0, an immediate return is effected.
Constraint: n0n0.
2:     ncolb – int64int32nag_int scalar
Default: The second dimension of the array b.
ncolbncolb, the number of columns of the matrix BB.
If ncolb = 0ncolb=0, the array b is not referenced.
Constraint: ncolb0ncolb0.

Input Parameters Omitted from the MATLAB Interface

lda ldb ldq

Output Parameters

1:     a(lda, : :) – double array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
If wantp = truewantp=true, the nn by nn part of a will contain the nn by nn orthogonal matrix PTPT, otherwise the nn by nn upper triangular part of a is used as internal workspace, but the strictly lower triangular part of a is not referenced.
2:     b(ldb, : :) – double array
The first dimension, ldb, of the array b will be
  • if ncolb > 0ncolb>0, ldbmax (1,n)ldbmax(1,n);
  • otherwise ldb1ldb1.
The second dimension of the array will be max (1,ncolb)max(1,ncolb)
The leading nn by ncolbncolb part of the array b stores the matrix QTBQTB.
3:     q(ldq, : :) – double array
The first dimension, ldq, of the array q will be
  • if wantq = truewantq=true, ldqmax (1,n)ldqmax(1,n);
  • otherwise ldq1ldq1.
The second dimension of the array will be max (1,n)max(1,n) if wantq = truewantq=true, and at least 11 otherwise
With wantq = truewantq=true, the leading nn by nn part of the array q will contain the orthogonal matrix QQ. Otherwise the array q is not referenced.
4:     sv(n) – double array
The array sv will contain the nn diagonal elements of the matrix SS.
5:     work( : :) – double array
Note: the dimension of the array work must be at least max (1,2 × (n1))max(1,2×(n-1)) if ncolb = 0ncolb=0 and wantq = falsewantq=false and wantp = falsewantp=false, max (1,3 × (n1))max(1,3×(n-1)) if (ncolb = 0ncolb=0 and wantq = falsewantq=false and wantp = truewantp=true) or (wantp = falsewantp=false and (ncolb > 0ncolb>0 or wantq = truewantq=true)), and at least max (1,5 × (n1))max(1,5×(n-1)) otherwise.
work(n)workn contains the total number of iterations taken by the QRQR algorithm.
The rest of the array is used as internal workspace.
6:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

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

  ifail = 1ifail=-1
On entry,n < 0n<0,
orlda < nlda<n,
orncolb < 0ncolb<0,
orldb < nldb<n and ncolb > 0ncolb>0,
orldq < nldq<n and wantq = truewantq=true.
W ifail > 0ifail>0
The QRQR algorithm has failed to converge in 50 × n50×n iterations. In this case sv(1),sv(2),,sv(ifail)sv1,sv2,,svifail may not have been found correctly and the remaining singular values may not be the smallest. The matrix RR will nevertheless have been factorized as R = QEPTR=QEPT, where EE is a bidiagonal matrix with sv(1),sv(2),,sv(n)sv1,sv2,,svn as the diagonal elements and work(1),work(2),,work(n1)work1,work2,,workn-1 as the superdiagonal elements.
This failure is not likely to occur.

Accuracy

The computed factors QQ, SS and PP satisfy the relation
QSPT = R + E,
QSPT=R+E,
where
Ecε A ,
Ecε A ,
εε is the machine precision, cc is a modest function of nn and . . denotes the spectral (two) norm. Note that A = sv(1)A=sv1.
A similar result holds for the computed matrix QTBQTB.
The computed matrix QQ satisfies the relation
Q = T + F,
Q=T+F,
where TT is exactly orthogonal and
Fdε,
Fdε,
where dd is a modest function of nn. A similar result holds for PP.

Further Comments

For given values of ncolb, wantq and wantp, the number of floating point operations required is approximately proportional to n3n3.
>Following the use of nag_eigen_real_triang_svd (f02wu) the rank of RR may be estimated as follows:
tol = eps;
irank = 1;
while (irank <= numel(sv) && sv(irank) >= tol*sv(1) )
  irank = irank + 1;
end
returns the value kk in irank, where kk is the smallest integer for which sv(k) < tol × sv(1)svk<tol×sv1, where toltol is typically the machine precision, so that irank is an estimate of the rank of SS and thus also of RR.

Example

function nag_eigen_real_triang_svd_example
a = [-4, -2, -3;
     0, -3, -2;
     0, 0, -4];
b = [-1; -1; -1];
wantq = true;
wantp = true;
[aOut, bOut, q, sv, work, ifail] = nag_eigen_real_triang_svd(a, b, wantq, wantp)
 

aOut =

   -0.4694   -0.4324   -0.7699
    0.7845    0.1961   -0.5883
   -0.4054    0.8801   -0.2471


bOut =

   -1.6716
   -0.3922
    0.2276


q =

    0.7699   -0.5883    0.2471
    0.4324    0.1961   -0.8801
    0.4694    0.7845    0.4054


sv =

    6.5616
    3.0000
    2.4384


work =

         0
         0
    1.0000
    0.5547
         0
   -0.8321
         0
         0
         0
         0


ifail =

                    0


function f02wu_example
a = [-4, -2, -3;
     0, -3, -2;
     0, 0, -4];
b = [-1; -1; -1];
wantq = true;
wantp = true;
[aOut, bOut, q, sv, work, ifail] = f02wu(a, b, wantq, wantp)
 

aOut =

   -0.4694   -0.4324   -0.7699
    0.7845    0.1961   -0.5883
   -0.4054    0.8801   -0.2471


bOut =

   -1.6716
   -0.3922
    0.2276


q =

    0.7699   -0.5883    0.2471
    0.4324    0.1961   -0.8801
    0.4694    0.7845    0.4054


sv =

    6.5616
    3.0000
    2.4384


work =

         0
         0
    1.0000
    0.5547
         0
   -0.8321
         0
         0
         0
         0


ifail =

                    0



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