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NAG[f08avc] NAG[nag_zgelqf] -  factorization of complex general rectangular matrix
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Calling Sequence
f08avc(a, tau, 'm'=m, 'n'=n, 'fail'=fail)
nag_zgelqf(. . .)
Parameters
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a - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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The diagonal elements of  are real.
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tau - Vector(1..dim, datatype=complex[8]);
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Note: the dimension, dim, of the array tau must be at least  .
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On exit: further details of the unitary matrix  .
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'm'=m - integer; (optional)
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Default value: the first dimension of the array a.
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On entry:  , the number of rows of the matrix  .
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Constraint:  . .
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'n'=n - integer; (optional)
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Default value: the second dimension of the array a.
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On entry:  , the number of columns of the matrix  .
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Constraint:  . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_zgelqf (f08avc) computes the  factorization of a complex  by  matrix.
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Error Indicators and Warnings
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"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.
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Accuracy
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The computed factorization is the exact factorization of a nearby matrix  , where
 = O(epsilon)*LinearAlgebra[Norm](A, 2)](/support/helpjp/helpview.aspx?si=10057/file08230/math296.png)
and  is the machine precision.
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Further Comments
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The total number of real floating-point operations is approximately  if  or  if  .
To form the unitary matrix  nag_zgelqf (f08avc) may be followed by a call to f08awc (nag_zunglq):
nag_zunglq (min(m,n),a,tau,'m'=n,'n'=n)
but note that the first dimension of the array a must be at least n, which may be larger than was required by nag_zgelqf (f08avc).
When  , it is often only the first  rows of  that are required, and they may be formed by the call:
nag_zunglq (m,a,tau,'m'=m,'n'=n)
To apply  to an arbitrary complex rectangular matrix  , nag_zgelqf (f08avc) may be followed by a call to f08axc (nag_zunmlq). For example,
nag_zunmlq ("Nag_LeftSide","Nag_ConjTrans",min(m,n),a,tau,c,'m'=m,'n'=p)
forms the matrix product  , where  is  by  .
The real analogue of this function is f08ahc (nag_dgelqf).
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Examples
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>
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m := 3:
n := 4:
a := Matrix([[0.28 -0.36*I , 0.5 -0.86*I , -0.77 -0.48*I , 1.58 +0.66*I ], [-0.5 -1.1*I , -1.21 +0.76*I , -0.32 -0.24*I , -0.27 -1.15*I ], [0.36 -0.51*I , -0.07000000000000001 +1.33*I , -0.75 +0.47*I , -0.08 +1.01*I ]], datatype=complex[8]):
tau := Vector(3, datatype=complex[8]):
NAG:-f08avc(a, tau, 'm' = m, 'n' = n):
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