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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 n n  by n n  upper triangular matrix R R  is factorized as
R = QSPT,
R=QSPT,
where Q Q  and P P  are n n  by n n  orthogonal matrices and S S  is an n n  by n n  diagonal matrix with non-negative diagonal elements, σ1,σ2,,σn σ1,σ2,,σn , ordered such that
σ1σ2σn0.
σ1σ2σn0.
The columns of Q Q  are the left-hand singular vectors of R R , the diagonal elements of S S  are the singular values of R R  and the columns of P P  are the right-hand singular vectors of R R .
Either or both of Q Q  and PT PT  may be requested and the matrix C C  given by
C = QTB,
C=QTB,
where B B  is an n n  by ncolb ncolb  given matrix, may also be requested.
The function obtains the singular value decomposition by first reducing R R  to bidiagonal form by means of Givens plane rotations and then using the QR QR  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 K K  is any orthogonal diagonal matrix so that
KKT = I
KKT=I
(that is the diagonal elements of K K  are + 1 +1  or 1 -1 ) then
A = (QK)S(PK)T
A= (QK) S (PK) T
is also a singular value decomposition of A A .

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 n n  by n n  upper triangular part of the array a must contain the upper triangular matrix R R .
2:     b(ldb, : : ) – double array
The first dimension, ldb, of the array b must satisfy
  • if ncolb > 0 ncolb>0 , ldbmax (1,n) ldbmax (1,n) ;
  • otherwise ldb1 ldb1 .
The second dimension of the array must be at least max (1,ncolb) max (1,ncolb)
With ncolb > 0 ncolb>0 , the leading n n  by ncolb ncolb  part of the array b must contain the matrix to be transformed.
3:     wantq – logical scalar
Must be true if the matrix Q Q  is required.
If wantq = false wantq=false , the array q is not referenced.
4:     wantp – logical scalar
Must be true if the matrix PT PT  is required, in which case PT PT  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.
n n , the order of the matrix R R .
If n = 0 n=0 , an immediate return is effected.
Constraint: n0 n0 .
2:     ncolb – int64int32nag_int scalar
Default: The second dimension of the array b.
ncolb ncolb , the number of columns of the matrix B B .
If ncolb = 0 ncolb=0 , the array b is not referenced.
Constraint: ncolb0 ncolb0 .

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 = true wantp=true , the n n by n n part of a will contain the n n by n n orthogonal matrix PT PT , otherwise the n n by n n 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 > 0 ncolb>0 , ldbmax (1,n) ldbmax (1,n) ;
  • otherwise ldb1 ldb1 .
The second dimension of the array will be max (1,ncolb) max (1,ncolb)
The leading n n by ncolb ncolb part of the array b stores the matrix QTB QTB .
3:     q(ldq, : : ) – double array
The first dimension, ldq, of the array q will be
  • if wantq = true wantq=true , ldqmax (1,n) ldqmax (1,n) ;
  • otherwise ldq1 ldq1 .
The second dimension of the array will be max (1,n) max (1,n)  if wantq = true wantq=true , and at least 1 1  otherwise
With wantq = true wantq=true , the leading n n by n n part of the array q will contain the orthogonal matrix Q Q . Otherwise the array q is not referenced.
4:     sv(n) – double array
The array sv will contain the n n diagonal elements of the matrix S S .
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 = 0 ncolb=0  and wantq = false wantq=false  and wantp = false wantp=false , max (1,3 × (n1)) max (1,3× (n-1) ) if (ncolb = 0 ncolb=0  and wantq = false wantq=false  and wantp = true wantp=true ) or (wantp = false wantp=false  and (ncolb > 0 ncolb>0  or wantq = true wantq=true )), and at least max (1,5 × (n1)) max (1,5× (n-1) ) otherwise.
work(n) work (n) contains the total number of iterations taken by the QR QR algorithm.
The rest of the array is used as internal workspace.
6:     ifail – int64int32nag_int scalar
ifail = 0 ifail=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 = 1 ifail=-1
On entry,n < 0 n<0 ,
orlda < n lda<n ,
orncolb < 0 ncolb<0 ,
orldb < n ldb<n  and ncolb > 0 ncolb>0 ,
orldq < n ldq<n  and wantq = true wantq=true .
W ifail > 0 ifail>0
The QR QR  algorithm has failed to converge in 50 × n 50×n  iterations. In this case sv(1),sv(2),,sv(ifail) sv (1) ,sv (2) ,,sv (ifail)  may not have been found correctly and the remaining singular values may not be the smallest. The matrix R R  will nevertheless have been factorized as R = QEPT R=QEPT , where E E  is a bidiagonal matrix with sv(1),sv(2),,sv(n) sv (1) ,sv (2) ,,sv (n)  as the diagonal elements and work(1),work(2),,work(n1) work (1) ,work (2) ,,work (n-1)  as the superdiagonal elements.
This failure is not likely to occur.

Accuracy

The computed factors Q Q , S S  and P P  satisfy the relation
QSPT = R + E,
QSPT=R+E,
where
Ecε A ,
E cε A ,
ε ε  is the machine precision, c c  is a modest function of n n  and . .  denotes the spectral (two) norm. Note that A = sv(1) A =sv (1) .
A similar result holds for the computed matrix QTB QTB .
The computed matrix Q Q  satisfies the relation
Q = T + F,
Q=T+F,
where T T  is exactly orthogonal and
Fdε,
F dε,
where d d  is a modest function of n n . A similar result holds for P P .

Further Comments

For given values of ncolb, wantq and wantp, the number of floating point operations required is approximately proportional to n3 n3 .
>Following the use of nag_eigen_real_triang_svd (f02wu) the rank of R R  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 k k  in irank, where k k  is the smallest integer for which sv(k) < tol × sv(1) sv (k) <tol×sv (1) , where tol tol  is typically the machine precision, so that irank is an estimate of the rank of S S  and thus also of R R .

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



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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