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NAG[f08ahc] NAG[nag_dgelqf] -  factorization of real general rectangular matrix
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Calling Sequence
f08ahc(a, tau, 'm'=m, 'n'=n, 'fail'=fail)
nag_dgelqf(. . .)
Parameters
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a - Matrix(1..dim1, 1..dim2, datatype=float[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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tau - Vector(1..dim, datatype=float[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 orthogonal 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_dgelqf (f08ahc) computes the  factorization of a real  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=10060/file08224/math292.png)
and  is the machine precision.
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Further Comments
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The total number of floating-point operations is approximately  if  or  if  .
To form the orthogonal matrix  nag_dgelqf (f08ahc) may be followed by a call to f08ajc (nag_dorglq):
nag_dorglq (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_dgelqf (f08ahc).
When  , it is often only the first  rows of  that are required, and they may be formed by the call:
nag_dorglq (m,a,tau,'m'=m,'n'=n)
To apply  to an arbitrary real rectangular matrix  , nag_dgelqf (f08ahc) may be followed by a call to f08akc (nag_dormlq). For example,
nag_dormlq ("Nag_LeftSide","Nag_Trans",min(m,n),a,tau,c,'m'=m,'n'=p)
forms the matrix product  , where  is  by  .
The complex analogue of this function is f08avc (nag_zgelqf).
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Examples
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m := 4:
n := 6:
a := Matrix([[-5.42, 3.28, -3.68, 0.27, 2.06, 0.46], [-1.65, -3.4, -3.2, -1.03, -4.06, -0.01], [-0.37, 2.35, 1.9, 4.31, -1.76, 1.13], [-3.15, -0.11, 1.99, -2.7, 0.26, 4.5]], datatype=float[8]):
tau := Vector(4, datatype=float[8]):
NAG:-f08ahc(a, tau, 'm' = m, 'n' = n):
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