◆ Zhbev()

Sub Zhbev	(	Jobz As	String,
		Uplo As	String,
		N As	Long,
		Kd As	Long,
		Ab() As	Complex,
		W() As	Double,
		Z() As	Complex,
		Info As	Long
	)

(Simple driver) Eigenvalues and eigenvectors of a Hermitian band matrix

Purpose: This routine computes all the eigenvalues and, optionally, eigenvectors of a Hermitian band matrix A.

Parameters

[in]	Jobz	= "N": Compute eigenvalues only. = "V": Compute eigenvalues and eigenvectors.
[in]	Uplo	= "U": Upper triangle of A is stored. = "L": Lower triangle of A is stored.
[in]	N	Order of the matrix A. (N >= 0) (If N = 0, returns without computation)
[in]	Kd	Number of super-diagonals of the matrix A if Uplo = "U" or number of sub-diagonals if Uplo = "L". (Kd >= 0)
[in,out]	Ab()	Array Ab(LAb1 - 1, LAb2 - 1) (LAb1 >= Kd + 1, LAb2 >= N) [in] N x N Hermitian band matrix A in Kd+1 x N symmetric band matrix form. Upper or lower part is to be stored in accordance with Uplo. [out] Ab() is overwritten by values generated during the reduction to tridiagonal form. Uplo = "U": The first super-diagonal and the diagonal of the tridiagonal matrix T are returned in rows Kd and Kd+1 of Ab(). Uplo = "L": The diagonal and first sub-diagonal of T are returned in the first two rows of Ab().
[out]	W()	Array W(LW - 1) (LW >= N) If Info = 0, the eigenvalues in ascending order.
[out]	Z()	Array Z(LZ1 - 1, LZ2 - 1) (LZ1 >= N, LZ2 >= N) Jobz = "V": If Info = 0, Z() contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z() holding the eigenvector associated with W(i). Jobz = "N": Z() is not referenced.
[out]	Info	= 0: Successful exit. = -1: The argument Jobz had an illegal value. (Jobz <> "V" nor "N") = -2: The argument Uplo had an illegal value. (Uplo <> "U" nor "L") = -3: The argument N had an illegal value. (N < 0) = -4: The argument Kd had an illegal value. (Kd < 0) = -5: The argument Ab() is invalid. = -6: The argument W() is invalid. = -7: The argument Z() is invalid. = i > 0: The algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.

Reference: LAPACK

Example Program: Compute all eigenvalues and eigenvectors of the Hermitian band matrix A, where
( 2.20 -0.32-0.81i 0 )

A = ( -0.32+0.81i 2.11 0.37+0.80i )

( 0 0.37-0.80i 2.93 )

Sub Ex_Zhbev()

Const N = 3, Kd = 1

Dim Ab(Kd, N - 1) As Complex, W(N - 1) As Double, Z(N - 1, N - 1) As Complex

Dim Info As Long

Ab(0, 0) = Cmplx(2.2, 0): Ab(0, 1) = Cmplx(2.11, 0): Ab(0, 2) = Cmplx(2.93, 0)

Ab(1, 0) = Cmplx(-0.32, 0.81): Ab(1, 1) = Cmplx(0.37, -0.8)

Call Zhbev("V", "L", N, Kd, Ab(), W(), Z(), Info)

Debug.Print "Eigenvalues =", W(0), W(1), W(2)

Debug.Print "Eigenvectors ="

Debug.Print Creal(Z(0, 0)), Cimag(Z(0, 0)), Creal(Z(0, 1)), Cimag(Z(0, 1))

Debug.Print Creal(Z(1, 0)), Cimag(Z(1, 0)), Creal(Z(1, 1)), Cimag(Z(1, 1))

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

Debug.Print Creal(Z(0, 2)), Cimag(Z(0, 2))

Debug.Print Creal(Z(1, 2)), Cimag(Z(1, 2))

Debug.Print Creal(Z(2, 2)), Cimag(Z(2, 2))

Debug.Print "Info =", Info

End Sub

Example Results: Eigenvalues = 1.04283948355918 2.52504447979701 3.67211603664382

Eigenvectors =

-0.563395195782213 0 0.745522345299636 0

-0.275043343741226 0.696203463844978 -0.102234588437164 0.258781301981572

-0.241207215775279 -0.253094504921304 -0.417819329339414 -0.438410500970378

-0.356065002532476 -0

0.221139195914327 -0.559758589658141

-0.493164614651588 -0.51746898859873

Info = 0