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NAG[f07brc] NAG[nag_zgbtrf] - factorization of complex by band matrix

Calling Sequence

f07brc(kl, ku, ab, ipiv, 'm'=m, 'n'=n, 'fail'=fail)

nag_zgbtrf(. . .)

Parameters

kl - integer;

On entry: , the number of subdiagonals within the band of the matrix .

Constraint: . .

ku - integer;

On entry: , the number of superdiagonals within the band of the matrix .

Constraint: . .

ab - 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 . This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements , for and , depends on the storage order argument as follows:

if , is stored as ;

if , is stored as .

On exit: ab is overwritten by details of the factorization. The elements, , of the upper triangular band factor with super-diagonals, and the multipliers, , used to form the lower triangular factor are stored. The elements , for and , and , for and , are stored using the same storage scheme as described for on entry.

ipiv - Vector(1..dim, datatype=integer[kernelopts('wordsize')/8]);

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

On exit: the pivot indices. Row of the matrix was interchanged with row , for .

'm'=m - integer; (optional)

Default value: the first dimension of the array ab.

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

Constraint: . .

'n'=n - integer; (optional)

Default value: the second dimension of the array ab.

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_zgbtrf (f07brc) computes the factorization of a complex by band matrix.

Description

nag_zgbtrf (f07brc) forms the factorization of a complex by band matrix using partial pivoting, with row interchanges. Usually , and then, if has non-zero subdiagonals and non-zero superdiagonals, the factorization has the form , where is a permutation matrix, is a lower triangular matrix with unit diagonal elements and at most non-zero elements in each column, and is an upper triangular band matrix with superdiagonals.

Note that is not a band matrix, but the non-zero elements of can be stored in the same space as the subdiagonal elements of . is a band matrix but with additional superdiagonals compared with . These additional superdiagonals are created by the row interchanges.

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: .

"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.

"NE_SINGULAR"

The factor is exactly singular.

	Accuracy
	The computed factors and are the exact factors of a perturbed matrix , where is a modest linear function of , and is the machine precision. This assumes .

Further Comments

The total number of real floating-point operations varies between approximately and , depending on the interchanges, assuming and .

A call to nag_zgbtrf (f07brc) may be followed by calls to the functions:

f07bsc (nag_zgbtrs) to solve , or ;

f07buc (nag_zgbcon) to estimate the condition number of .

The real analogue of this function is f07bdc (nag_dgbtrf).

Examples

m := 4:
n := 4:
kl := 1:
ku := 2:
ab := Matrix([[0 +0*I , -1.65 +2.26*I , -2.05 -0.85*I , 0.97 -2.84*I , 0 +0*I ], [0 +6.3*I , -1.48 -1.75*I , -3.99 +4.01*I , 0.59 -0.48*I , 0 +0*I ], [-0.77 +2.83*I , -1.06 +1.94*I , 3.33 -1.04*I , 0 +0*I , 0 +0*I ], [4.48 -1.09*I , -0.46 -1.72*I , 0 +0*I , 0 +0*I , 0 +0*I ]], datatype=complex[8], order='C_order'):
ipiv := Vector(4, datatype=integer[kernelopts('wordsize')/8]):
NAG:-f07brc(kl, ku, ab, ipiv, 'm' = m, 'n' = n):

ab;

ipiv;

	See Also
	Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore f07 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.