◆ Zhetrd()

Sub Zhetrd	(	Uplo As	String,
		N As	Long,
		A() As	Complex,
		D() As	Double,
		E() As	Double,
		Tau() As	Complex,
		Info As	Long
	)

Reduces a real Hermitian matrix to tridiagonal form

Purpose: This routine reduces a Hermitian matrix A to real symmetric tridiagonal form T by an unitary similarity transformation: Q^H * A * Q = T.

Parameters

[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,out]	A()	Array A(LA1 - 1, LA2 - 1) (LA1 >= N, LA2 >= N) [in] The Hermitian matrix A. If Uplo = "U", the leading N x N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If Uplo = "L", the leading N x N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. [out] If Uplo = "U", the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array Tau(), represent the unitary matrix Q as a product of elementary reflectors; If Uplo = "L", the diagonal and first subdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array Tau(), represent the unitary matrix Q as a product of elementary reflectors. See Further Details.
[out]	D()	Array D(LD - 1) (LD >= N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i, i).
[out]	E()	Array E(LE - 1) (LE >= N - 1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i, i+1) if Uplo = "U", E(i) = A(i+1, i) if Uplo = "L".
[out]	Tau()	Array Tau(LTau - 1) (LTau >= N - 1) The scalar factors of the elementary reflectors (see Further Details).
[out]	Info	= 0: Successful exit. = -1: The argument Uplo had an illegal value. (Uplo <> "U" nor "L") = -2: The argument N had an illegal value. (N < 0) = -3: The argument A() is invalid. = -4: The argument D() is invalid. = -5: The argument E() is invalid. = -6: The argument Tau() is invalid.

Further Details: If Uplo = "U", the matrix Q is represented as a product of elementary reflectors
Q = H(N-1) . . . H(2) H(1).

Each H(i) has the form
H(i) = I - tau*v*v^H

where tau is a complex scalar, and v is a complex vector with v(i+1〜N) = 0 and v(i) = 1. v(1〜i-1) is stored on exit in A(0〜i-2, i), and tau in Tau(i-1).

If Uplo = "L", the matrix Q is represented as a product of elementary reflectors
Q = H(1) H(2) . . . H(N-1).

Each H(i) has the form
H(i) = I - tau*v*v^H

where tau is a complex scalar, and v is a complex vector with v(1〜i) = 0 and v(i+1) = 1; v(i+2〜N) is stored on exit in A(i+1〜N-1, i-1), and tau in Tau(i-1).

The contents of A() on exit are illustrated by the following examples with N = 5:
if Uplo = "L": if Uplo = "U":

( d e v3 v4 v5 ) ( d )

( d e v4 v5 ) ( e d )

( d e v5 ) ( v1 e d )

( d e ) ( v1 v2 e d )

( d ) ( v1 v2 v3 e d )

where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector v defining H(i).

Reference: LAPACK

Example Program: Compute all eigenvalues of the Hermitian matrix A, where
( 0.20 -0.11+0.93i 0.81-0.37i )

A = ( -0.11-0.93i -0.32 -0.80+0.92i )

( 0.81+0.37i -0.80-0.92i -0.29 )

Reduces to real tridiagonal form by Zhetrd, then Dsterf is applied.
Sub Ex_Zhetrd_Dsterf()

Const N = 3

Dim A(N - 1, N - 1) As Complex, Info As Long

Dim D(N - 1) As Double, E(N - 2) As Double, Tau(N - 2) As Complex

A(0, 0) = Cmplx(0.2, 0)

A(1, 0) = Cmplx(-0.11, -0.93): A(1, 1) = Cmplx(-0.32, 0)

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

Call Zhetrd("L", N, A(), D(), E(), Tau(), Info)

If Info <> 0 Then

Debug.Print "Error in Zhetrd: Info =", Info

Exit Sub

End If

Call Dsterf(N, D(), E(), Info)

If Info <> 0 Then

Debug.Print "Error in Dsterf: Info =", Info

Exit Sub

End If

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

End Sub

Example Results: Eigenvalues = -2.05348849668514 0.124622388617308 1.51886610806783