Unitary reduction of complex general rectangular matrix to bidiagonal form

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      - f08aec (nag_dgeqrf)
      - f08afc (nag_dorgqr)
      - f08agc (nag_dormqr)
      - f08ahc (nag_dgelqf)
      - f08ajc (nag_dorglq)
      - f08akc (nag_dormlq)
      - f08asc (nag_zgeqrf)
      - f08atc (nag_zungqr)
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      - f08awc (nag_zunglq)
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      - f08fcc (nag_dsyevd)
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      - f08fqc (nag_zheevd)
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      - f08ftc (nag_zungtr)
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      - f08gsc (nag_zhptrd)
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      - f08guc (nag_zupmtr)
      - f08hcc (nag_dsbevd)
      - f08hec (nag_dsbtrd)
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      - f08lsc (nag_zgbbrd)
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      - f08njc (nag_dgebak)
      - f08nsc (nag_zgehrd)
      - f08ntc (nag_zunghr)
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      - f08nvc (nag_zgebal)
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      - f08wsc (nag_zgghrd)
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NAG[f08ksc] NAG[nag_zgebrd] - Unitary reduction of complex general rectangular matrix to bidiagonal form

Calling Sequence

f08ksc(a, d, e, tauq, taup, 'm'=m, 'n'=n, 'fail'=fail)

nag_zgebrd(. . .)

Parameters

a - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

On entry: the by matrix .

On exit: if , the diagonal and first superdiagonal are overwritten by the upper bidiagonal matrix , elements below the diagonal are overwritten by details of the unitary matrix and elements above the first superdiagonal are overwritten by details of the unitary matrix .

If , the diagonal and first subdiagonal are overwritten by the lower bidiagonal matrix , elements below the first subdiagonal are overwritten by details of the unitary matrix and elements above the diagonal are overwritten by details of the unitary matrix .

d - Vector(1..dim, datatype=float[8]);

Note: the dimension, dim, of the array d must be at least .

On exit: the diagonal elements of the bidiagonal matrix .

e - Vector(1..dim, datatype=float[8]);

Note: the dimension, dim, of the array e must be at least .

On exit: the off-diagonal elements of the bidiagonal matrix .

tauq - Vector(1..dim, datatype=complex[8]);

Note: the dimension, dim, of the array tauq must be at least .

On exit: further details of the matrix .

taup - Vector(1..dim, datatype=complex[8]);

Note: the dimension, dim, of the array taup must be at least .

On exit: further details of the matrix .

'm'=m - integer; (optional)

Default value: the first dimension of the array a.

On entry: , the number of rows of the matrix .

Constraint: . .

'n'=n - integer; (optional)

Default value: the second dimension of the array a.

On entry: , the number of columns of the matrix .

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_zgebrd (f08ksc) reduces a complex by matrix to bidiagonal form.

Description

nag_zgebrd (f08ksc) reduces a complex by matrix to real bidiagonal form by a unitary transformation: , where and are unitary matrices of order and respectively.

If , the reduction is given by:

A = Q*binomial(B[1], 0)*P^H and Q*binomial(B[1], 0)*P^H = Q[1]*B[1]*P^H

where is a real by upper bidiagonal matrix and consists of the first columns of .

If , the reduction is given by

where is a real by lower bidiagonal matrix and consists of the first rows of .

The unitary matrices and 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 and in this representation (see Section [Further Comments]).

	Error Indicators and Warnings
	"NE_ALLOC_FAIL" Dynamic memory allocation failed. "NE_BAD_PARAM" On entry, argument had an illegal value. "NE_INT" On entry, . Constraint: . On entry, . Constraint: . "NE_INTERNAL_ERROR" An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.

	Accuracy
	The computed bidiagonal form satisfies , where is a modestly increasing function of , and is the machine precision. The elements of themselves may be sensitive to small perturbations in 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 if or if .

If , it can be more efficient to first call f08asc (nag_zgeqrf) to perform a factorization of , and then to call nag_zgebrd (f08ksc) to reduce the factor to bidiagonal form. This requires approximately floating-point operations.

If , it can be more efficient to first call f08avc (nag_zgelqf) to perform an factorization of , and then to call nag_zgebrd (f08ksc) to reduce the factor to bidiagonal form. This requires approximately operations.

To form the unitary matrices and/or nag_zgebrd (f08ksc) may be followed by calls to f08ktc (nag_zungbr):

to form the by unitary matrix

nag_zungbr ("Nag_FormQ",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_zgebrd (f08ksc);

To apply or to a complex rectangular matrix , nag_zgebrd (f08ksc) may be followed by a call to f08kuc (nag_zunmbr).

The real analogue of this function is f08kec (nag_dgebrd).

Examples

m := 6:
n := 4:
a := Matrix([[0.96 -0.8100000000000001*I , -0.03 +0.96*I , -0.91 +2.06*I , -0.05 +0.41*I ], [-0.98 +1.98*I , -1.2 +0.19*I , -0.66 +0.42*I , -0.8100000000000001 +0.5600000000000001*I ], [0.62 -0.46*I , 1.01 +0.02*I , 0.63 -0.17*I , -1.11 +0.6*I ], [-0.37 +0.38*I , 0.19 -0.54*I , -0.98 -0.36*I , 0.22 -0.2*I ], [0.83 +0.51*I , 0.2 +0.01*I , -0.17 -0.46*I , 1.47 +1.59*I ], [1.08 -0.28*I , 0.2 -0.12*I , -0.07000000000000001 +1.23*I , 0.26 +0.26*I ]], datatype=complex[8]):
d := Vector(4, datatype=float[8]):
e := Vector(3, datatype=float[8]):
tauq := Vector(4, datatype=complex[8]):
taup := Vector(4, datatype=complex[8]):
NAG:-f08ksc(a, d, e, tauq, taup, 'm' = m, 'n' = n):

tauq;

taup;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f08 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.