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NAG Toolbox: nag_lapack_dgebrd (f08ke)

Purpose

nag_lapack_dgebrd (f08ke) reduces a real mm by nn matrix to bidiagonal form.

Syntax

[a, d, e, tauq, taup, info] = f08ke(a, 'm', m, 'n', n)
[a, d, e, tauq, taup, info] = nag_lapack_dgebrd(a, 'm', m, 'n', n)

Description

nag_lapack_dgebrd (f08ke) reduces a real mm by nn matrix AA to bidiagonal form BB by an orthogonal transformation: A = QBPTA=QBPT, where QQ and PTPT are orthogonal matrices of order mm and nn respectively.
If mnmn, the reduction is given by:
A = Q
(B1)
0
PT = Q1 B1 PT ,
A =Q B1 0 PT = Q1 B1 PT ,
where B1B1 is an nn by nn upper bidiagonal matrix and Q1Q1 consists of the first nn columns of QQ.
If m < nm<n, the reduction is given by
A = Q
(B10)
PT = Q B1 P1T ,
A =Q B1 0 PT = Q B1 P1T ,
where B1B1 is an mm by mm lower bidiagonal matrix and P1TP1T consists of the first mm rows of PTPT.
The orthogonal matrices QQ and PP are not formed explicitly but are represented as products of elementary reflectors (see the F08 Chapter Introduction for details). Functions are provided to work with QQ and PP in this representation (see Section [Further Comments]).

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1:     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: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda work 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).
If mnmn, the diagonal and first superdiagonal store the upper bidiagonal matrix BB, elements below the diagonal store details of the orthogonal matrix QQ and elements above the first superdiagonal store details of the orthogonal matrix PP.
If m < nm<n, the diagonal and first subdiagonal store the lower bidiagonal matrix BB, elements below the first subdiagonal store details of the orthogonal matrix QQ and elements above the diagonal store details of the orthogonal matrix PP.
2:     d( : :) – double array
Note: the dimension of the array d must be at least max (1,min (m,n))max(1,min(m,n)).
The diagonal elements of the bidiagonal matrix BB.
3:     e( : :) – double array
Note: the dimension of the array e must be at least max (1,min (m,n)1)max(1,min(m,n)-1).
The off-diagonal elements of the bidiagonal matrix BB.
4:     tauq( : :) – double array
Note: the dimension of the array tauq must be at least max (1,min (m,n))max(1,min(m,n)).
Further details of the orthogonal matrix QQ.
5:     taup( : :) – double array
Note: the dimension of the array taup must be at least max (1,min (m,n))max(1,min(m,n)).
Further details of the orthogonal matrix PP.
6:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: d, 6: e, 7: tauq, 8: taup, 9: work, 10: lwork, 11: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed bidiagonal form BB satisfies QBPT = A + EQBPT=A+E, where
E2 c (n) ε A2 ,
E2 c (n) ε A2 ,
c(n)c(n) is a modestly increasing function of nn, and εε is the machine precision.
The elements of BB themselves may be sensitive to small perturbations in AA or to rounding errors in the computation, but this does not affect the stability of the singular values and vectors.

Further Comments

The total number of floating point operations is approximately (4/3)n2(3mn)43n2(3m-n) if mnmn or (4/3)m2(3nm)43m2(3n-m) if m < nm<n.
If mnmn, it can be more efficient to first call nag_lapack_dgeqrf (f08ae) to perform a QRQR factorization of AA, and then to call nag_lapack_dgebrd (f08ke) to reduce the factor RR to bidiagonal form. This requires approximately 2n2(m + n)2n2(m+n) floating point operations.
If mnmn, it can be more efficient to first call nag_lapack_dgelqf (f08ah) to perform an LQLQ factorization of AA, and then to call nag_lapack_dgebrd (f08ke) to reduce the factor LL to bidiagonal form. This requires approximately 2m2(m + n)2m2(m+n) operations.
To form the orthogonal matrices PTPT and/or QQ nag_lapack_dgebrd (f08ke) may be followed by calls to nag_lapack_dorgbr (f08kf):
to form the mm by mm orthogonal matrix QQ 
[a, info] = f08kf('Q', k, a, tauq);
but note that the second dimension of the array a must be at least m, which may be larger than was required by nag_lapack_dgebrd (f08ke);
to form the nn by nn orthogonal matrix PTPT 
[a, info] = f08kf('P', k, a, taup);
but note that the first dimension of the array a, specified by the parameter lda, must be at least n, which may be larger than was required by nag_lapack_dgebrd (f08ke).
To apply QQ or PP to a real rectangular matrix CC, nag_lapack_dgebrd (f08ke) may be followed by a call to nag_lapack_dormbr (f08kg).
The complex analogue of this function is nag_lapack_zgebrd (f08ks).

Example

function nag_lapack_dgebrd_example
a = [-0.57, -1.28, -0.39, 0.25;
     -1.93, 1.08, -0.31, -2.14;
     2.3, 0.24, 0.4, -0.35;
     -1.93, 0.64, -0.66, 0.08;
     0.15, 0.3, 0.15, -2.13;
     -0.02, 1.03, -1.43, 0.5];
[aOut, d, e, tauq, taup, info] = nag_lapack_dgebrd(a)
 

aOut =

    3.6177    1.2587   -0.4668   -0.4110
    0.4609    2.4161    1.5262   -0.2095
   -0.5492    0.1219   -1.9213   -1.1895
    0.4609    0.0770    0.0557   -1.4265
   -0.0358    0.3309   -0.1248   -0.4048
    0.0048    0.2796    0.8322    0.2196


d =

    3.6177
    2.4161
   -1.9213
   -1.4265


e =

    1.2587
    1.5262
   -1.1895


tauq =

    1.1576
    1.6550
    1.1687
    1.6501


taup =

    1.4422
    1.9159
         0
         0


info =

                    0


function f08ke_example
a = [-0.57, -1.28, -0.39, 0.25;
     -1.93, 1.08, -0.31, -2.14;
     2.3, 0.24, 0.4, -0.35;
     -1.93, 0.64, -0.66, 0.08;
     0.15, 0.3, 0.15, -2.13;
     -0.02, 1.03, -1.43, 0.5];
[aOut, d, e, tauq, taup, info] = f08ke(a)
 

aOut =

    3.6177    1.2587   -0.4668   -0.4110
    0.4609    2.4161    1.5262   -0.2095
   -0.5492    0.1219   -1.9213   -1.1895
    0.4609    0.0770    0.0557   -1.4265
   -0.0358    0.3309   -0.1248   -0.4048
    0.0048    0.2796    0.8322    0.2196


d =

    3.6177
    2.4161
   -1.9213
   -1.4265


e =

    1.2587
    1.5262
   -1.1895


tauq =

    1.1576
    1.6550
    1.1687
    1.6501


taup =

    1.4422
    1.9159
         0
         0


info =

                    0



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