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NAG[f08awc] NAG[nag_zunglq] - Form all or part of unitary  from  factorization determined by f08avc (nag_zgelqf)
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Calling Sequence
f08awc(k, a, tau, 'm'=m, 'n'=n, 'fail'=fail)
nag_zunglq(. . .)
Parameters
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k - integer;
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On entry:  , the number of elementary reflectors whose product defines the matrix  .
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Constraint:  . .
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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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On entry: details of the vectors which define the elementary reflectors, as returned by f08avc (nag_zgelqf).
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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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'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_zunglq (f08awc) generates all or part of the complex unitary matrix  from an  factorization computed by f08avc (nag_zgelqf).
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Description
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nag_zunglq (f08awc) is intended to be used after a call to f08avc (nag_zgelqf), which performs an  factorization of a complex matrix  . The unitary matrix  is represented as a product of elementary reflectors.
This function may be used to generate  explicitly as a square matrix, or to form only its leading rows.
Usually  is determined from the  factorization of a  by  matrix  with  . The whole of  may be computed by:
nag_zunglq (p,a,tau,'m'=n,'n'=n)
(note that the array a must have at least  rows) or its leading  rows by:
nag_zunglq (p,a,tau,'m'=p,'n'=n)
The rows of  returned by the last call form an orthonormal basis for the space spanned by the rows of  ; thus f08avc (nag_zgelqf) followed by nag_zunglq (f08awc) can be used to orthogonalise the rows of  .
The information returned by the  factorization functions also yields the  factorization of the leading  rows of  , where  . The unitary matrix arising from this factorization can be computed by:
nag_zunglq (k,a,tau,'m'=n,'n'=n)
or its leading  rows by:
nag_zunglq (k,a,tau,'m'=k,'n'=n)
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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_INT_2"
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 matrix  differs from an exactly unitary matrix by a matrix  such that
 = O(epsilon)](/support/helpjp/helpview.aspx?si=10059/file08231/math292.png)
where  is the machine precision.
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Further Comments
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The total number of real floating-point operations is approximately  ; when  , the number is approximately  .
The real analogue of this function is f08ajc (nag_dorglq).
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Examples
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m := 3:
n := 4:
k := 3:
a := Matrix([[-2.225511177235469 -0*I , 0.2438442594612761 -0.3082069932914802*I , -0.2741364985191491 -0.230966496866882*I , 0.5807709514701289 +0.3468663602164287*I ], [0.8207552577961005 +1.238457046719376*I , 1.688100989346069 -0*I , -0.1936415258449461 +0.5429517855236941*I , 0.2789084851242076 -0.2203175797458332*I ], [0.001033470433007241 -0.6822252862760427*I , 0.774751330164006 -0.6154727158531147*I , -1.590258250459319 -0*I , -0.1267668516113225 +0.1109845202357172*I ]], datatype=complex[8]):
tau := Vector([1.12581379184436 +0.1617605895141772*I, 1.099053669168954 +0.5468590598466785*I, 1.132925657311579 -0.9590540478961487*I], datatype=complex[8]):
NAG:-f08awc(k, a, tau, 'm' = m, 'n' = n):
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