◆ WZpbsv2()

Function WZpbsv2	(	Uplo As	String,
		N As	Long,
		Kd As	Long,
		Ab As	Variant,
		B As	Variant,
		Optional Nrhs As	Long = `1`
	)

Solution to system of linear equations AX = B for a Hermitian positive definite band matrix (complex numbers in pairs of cells)

Purpose: WZpbsv2 computes the solution to a complex system of linear equations
A * X = B,

where A is an N x N Hermitian positive definite band matrix, and X and B are N x Nrhs matrices.

The Cholesky decomposition is used to factor A as
A = U^H * U, if Uplo = "U", or

A = L * L^H, if Uplo = "L",

where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of super-diagonals or sub-diagonals as A. The factored form of A is then used to solve the system of equations A * X = B.

To represent complex numbers, a real part and an imaginary part are stored in a pair of adjacent cells (a real part in a left cell, and an imaginary part in a right cell). The computed results are stored in the same way.

Returns

N+1 x 2Nrhs

	Column 1	Column 2	. . .	Column 2Nrhs
Rows 1 to N	Solution matrix X (a real part and an imaginary part are stored in a pair of adjacent columns (a real part is left and an imaginary part is right))
Row N+1	Reciprocal condition number	Return code	. . .	0

Return code.
= 0: Successful exit.
= i > 0: The leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been computed.

Parameters

[in]	Uplo	= "U": Upper triangle of A is stored. = "L": Lower triangle of A is stored.
[in]	N	Number of linear equations, i.e., order of the matrix A. (N >= 1)
[in]	Kd	Number of superdiagonals or subdiagonals of the matrix A. (Kd >= 0)
[in]	Ab	(Kd+1 x 2N) N x N coefficient matrix A. (Symmetric band matrix form. See below for details)
[in]	B	(N x 2Nrhs) N x Nrhs right hand side matrix B.
[in]	Nrhs	(Optional) Number of columns of right hand side matrix B. (Nrhs >= 1) (default = 1)

Further Details

The symmetric band matrix form is illustrated by the following example, when n = 6, kd = 2, and uplo = "U":

     *    *   a13  a24  a35  a46
     *   a12  a23  a34  a45  a56
    a11  a22  a33  a44  a55  a66

Similarly, if uplo = "L" the format of A is as follows:

    a11  a22  a33  a44  a55  a66
    a21  a32  a43  a54  a65   *
    a31  a42  a53  a64   *    *

Array elements marked * are not used by the routine.

Reference: LAPACK

Example: Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where A is an Hermitian positive definite band matrix and
( 2.88 0.29-0.44i 0 )

A = ( 0.29+0.44i 0.62 -0.01-0.02i )

( 0 -0.01+0.02i 0.46 )

( 1.6236-0.7300i )

B = ( 0.1581+0.1537i )

( 0.1132-0.2290i )

WZpbsv2