◆ Zgbtrf()

Sub Zgbtrf	(	M As	Long,
		N As	Long,
		Kl As	Long,
		Ku As	Long,
		Ab() As	Complex,
		IPiv() As	Long,
		Info As	Long
	)

LU factorization of a complex band matrix

Purpose: This routine computes an LU factorization of a complex m x n band matrix A using partial pivoting with row interchanges.
This is the blocked version of the algorithm calling Level 3 BLAS.

Parameters

[in]	M	Number of rows of the matrix A. (M >= 0) (If M = 0, returns without computation)
[in]	N	Number of columns of the matrix A. (N >= 0) (If N = 0, returns without computation)
[in]	Kl	Number of subdiagonals within the band of A (Kl >= 0)
[in]	Ku	Number of superdiagonals within the band of A (Ku >= 0)
[in,out]	Ab()	Array Ab(LAb1 - 1, LAb2 - 1) (LAb1 >= 2Kl + Ku + 1, LAb2 >= N) [in] The matrix A in band matrix form, in rows Kl+1 to 2Kl+Ku+1; rows 1 to Kl of the array need not be set. [out] Details of the factorization: U is stored as an upper triangular band matrix with kl+ku super-diagonals in rows 1 to Kl+Ku+1, and the multipliers used during the factorization are stored in rows Kl+Ku+2 to 2*Kl+Ku+1. See below for further details.
[out]	IPiv()	Array IPiv(LIPiv - 1) (LIPiv >= N) Pivot indices; for 1 <= i <= min(M, N), row i of the matrix was interchanged with row IPiv(i-1).
[out]	Info	= 0: Successful exit. = -1: The argument M had an illegal value. (M < 0) = -2: The argument N had an illegal value. (N < 0) = -3: The argument Kl had an illegal value. (Kl < 0) = -4: The argument Ku had an illegal value. (Ku < 0) = -5: The argument Ab() is invalid. = -6: The argument IPiv() is invalid. = i > 0: The i-th diagonal element of the factor U is exactly zero. The factorization has been completed, but the factor U is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Further Details

The band matrix form is illustrated by the following example, when M = N = 6, Kl = 2, Ku = 1:

On entry:
   *    *    *    +    +    +
   *    *    +    +    +    +
   *   a12  a23  a34  a45  a56
  a11  a22  a33  a44  a55  a66
  a21  a32  a43  a54  a65   *
  a31  a42  a53  a64   *    *

On exit:
   *    *    *   u14  u25  u36
   *    *   u13  u24  u35  u46
   *   u12  u23  u34  u45  u56
  u11  u22  u33  u44  u55  u66
  m21  m32  m43  m54  m65   *
  m31  m42  m53  m64   *    *

Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges.

Reference: LAPACK

Example Program: Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where
( 0.81+0.37i 0.20-0.11i 0 )

A = ( 0.64+0.51i -0.80-0.92i -0.93-0.32i )

( 0 0.71+0.59i -0.29+0.86i )

( -0.0484+0.2644i )

B = ( -0.2644-1.0228i )

( -0.5299+1.5025i )

Sub Ex_Zgbtrf()

Const N = 3, Kl = 1, Ku = 1

Dim Ab(2 * Kl + Ku, N - 1) As Complex, B(N - 1) As Complex, IPiv(N - 1) As Long

Dim ANorm As Double, RCond As Double, Info As Long

Ab(1, 1) = Cmplx(0.2, -0.11): Ab(1, 2) = Cmplx(-0.93, -0.32)

Ab(2, 0) = Cmplx(0.81, 0.37): Ab(2, 1) = Cmplx(-0.8, -0.92): Ab(2, 2) = Cmplx(-0.29, 0.86)

Ab(3, 0) = Cmplx(0.64, 0.51): Ab(3, 1) = Cmplx(0.71, 0.59)

B(0) = Cmplx(-0.0484, 0.2644): B(1) = Cmplx(-0.2644, -1.0228): B(2) = Cmplx(-0.5299, 1.5025)

ANorm = Zlangb("1", N, Kl, Ku, Ab(), , Kl)

Call Zgbtrf(N, N, Kl, Ku, Ab(), IPiv(), Info)

If Info = 0 Then Call Zgbtrs("N", N, Kl, Ku, Ab(), IPiv(), B(), Info)

If Info = 0 Then Call Zgbcon("1", N, Kl, Ku, Ab(), IPiv(), ANorm, RCond, Info)

Debug.Print "X =",

Debug.Print Creal(B(0)), Cimag(B(0)), Creal(B(1)), Cimag(B(1)), Creal(B(2)), Cimag(B(2))

Debug.Print "RCond =", RCond

Debug.Print "Info =", Info

End Sub

Example Results: X = -0.15 0.19 0.2 0.94 0.79 -0.13

RCond = 0.187722560135325

Info = 0