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NAG Toolbox: nag_lapack_zgebrd (f08ks)

Purpose

nag_lapack_zgebrd (f08ks) reduces a complex mm by nn matrix to bidiagonal form.

Syntax

[a, d, e, tauq, taup, info] = f08ks(a, 'm', m, 'n', n)
[a, d, e, tauq, taup, info] = nag_lapack_zgebrd(a, 'm', m, 'n', n)

Description

nag_lapack_zgebrd (f08ks) reduces a complex mm by nn matrix AA to real bidiagonal form BB by a unitary transformation: A = QBPHA=QBPH, where QQ and PHPH are unitary matrices of order mm and nn respectively.
If mnmn, the reduction is given by:
A = Q
(B1)
0
PH = Q1 B1 PH ,
A =Q B1 0 PH = Q1 B1 PH ,
where B1B1 is a real 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)
PH = Q B1 P1H ,
A =Q B1 0 PH = Q B1 P1H ,
where B1B1 is a real mm by mm lower bidiagonal matrix and P1HP1H consists of the first mm rows of PHPH.
The unitary 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, : :) – complex 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, : :) – complex 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 unitary matrix QQ and elements above the first superdiagonal store details of the unitary 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 unitary matrix QQ and elements above the diagonal store details of the unitary 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( : :) – complex 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 unitary matrix QQ.
5:     taup( : :) – complex 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 unitary 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 QBPH = A + EQBPH=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 real floating point operations is approximately 16n2(3mn) / 316n2(3m-n)/3 if mnmn or 16m2(3nm) / 316m2(3n-m)/3 if m < nm<n.
If mnmn, it can be more efficient to first call nag_lapack_zgeqrf (f08as) to perform a QRQR factorization of AA, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor RR to bidiagonal form. This requires approximately 8n2(m + n)8n2(m+n) floating point operations.
If mnmn, it can be more efficient to first call nag_lapack_zgelqf (f08av) to perform an LQLQ factorization of AA, and then to call nag_lapack_zgebrd (f08ks) to reduce the factor LL to bidiagonal form. This requires approximately 8m2(m + n)8m2(m+n) operations.
To form the unitary matrices PHPH and/or QQ nag_lapack_zgebrd (f08ks) may be followed by calls to nag_lapack_zungbr (f08kt):
to form the mm by mm unitary matrix QQ 
[a, info] = f08kt('Q', n, 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_zgebrd (f08ks);
to form the nn by nn unitary matrix PHPH 
[a, info] = f08kt('P', m, 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_zgebrd (f08ks).
To apply QQ or PP to a complex rectangular matrix CC, nag_lapack_zgebrd (f08ks) may be followed by a call to nag_lapack_zunmbr (f08ku).
The real analogue of this function is nag_lapack_dgebrd (f08ke).

Example

function nag_lapack_zgebrd_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
      -0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
      -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
      -0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
      0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.2 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
[aOut, d, e, tauq, taup, info] = nag_lapack_zgebrd(a)
 

aOut =

  -3.0870 + 0.0000i   2.1126 + 0.0000i   0.0543 + 0.4543i   0.3757 + 0.1070i
  -0.3270 + 0.4238i   2.0660 + 0.0000i   1.2628 + 0.0000i   0.0283 + 0.1650i
   0.1692 - 0.0798i  -0.2585 - 0.0137i   1.8731 + 0.0000i  -1.6126 + 0.0000i
  -0.1060 + 0.0727i   0.0582 + 0.0068i  -0.3219 + 0.3404i   2.0022 + 0.0000i
   0.1729 + 0.1606i   0.0884 - 0.1430i  -0.4052 - 0.2475i   0.2871 + 0.1826i
   0.2699 - 0.0152i  -0.0551 - 0.1065i   0.2172 + 0.2910i   0.5596 - 0.0569i


d =

   -3.0870
    2.0660
    1.8731
    2.0022


e =

    2.1126
    1.2628
   -1.6126


tauq =

   1.3110 - 0.2624i
   1.7965 - 0.0234i
   1.2420 - 0.1807i
   1.0144 + 0.6225i


taup =

   1.2312 - 0.5404i
   1.2623 - 0.9286i
   1.7829 - 0.6221i
   0.0000 + 0.0000i


info =

                    0


function f08ks_example
a = [ 0.96 - 0.81i,  -0.03 + 0.96i,  -0.91 + 2.06i,  -0.05 + 0.41i;
      -0.98 + 1.98i,  -1.2 + 0.19i,  -0.66 + 0.42i, ...
      -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i,  -1.11 + 0.6i;
      -0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.2i;
      0.83 + 0.51i,  0.2 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.2 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
[aOut, d, e, tauq, taup, info] = f08ks(a)
 

aOut =

  -3.0870 + 0.0000i   2.1126 + 0.0000i   0.0543 + 0.4543i   0.3757 + 0.1070i
  -0.3270 + 0.4238i   2.0660 + 0.0000i   1.2628 + 0.0000i   0.0283 + 0.1650i
   0.1692 - 0.0798i  -0.2585 - 0.0137i   1.8731 + 0.0000i  -1.6126 + 0.0000i
  -0.1060 + 0.0727i   0.0582 + 0.0068i  -0.3219 + 0.3404i   2.0022 + 0.0000i
   0.1729 + 0.1606i   0.0884 - 0.1430i  -0.4052 - 0.2475i   0.2871 + 0.1826i
   0.2699 - 0.0152i  -0.0551 - 0.1065i   0.2172 + 0.2910i   0.5596 - 0.0569i


d =

   -3.0870
    2.0660
    1.8731
    2.0022


e =

    2.1126
    1.2628
   -1.6126


tauq =

   1.3110 - 0.2624i
   1.7965 - 0.0234i
   1.2420 - 0.1807i
   1.0144 + 0.6225i


taup =

   1.2312 - 0.5404i
   1.2623 - 0.9286i
   1.7829 - 0.6221i
   0.0000 + 0.0000i


info =

                    0



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