◆ Zpttrf()

Sub Zpttrf	(	N As	Long,
		D() As	Double,
		E() As	Complex,
		Info As	Long
	)

LDL^H factorization of a Hermitian positive definite tridiagonal matrix

Purpose: This routine computes the L*D*L^H factorization of a Hermitian positive definite tridiagonal matrix A. The factorization may also be regarded as having the form A = U^H*D*U.

Parameters

[in]	N	Order of the matrix A. (N >= 0) (If N = 0, returns without computation)
[in,out]	D()	Array D(LD - 1) (LD >= N) [in] N diagonal elements of the Hermitian positive definite tridiagonal matrix A. [out] N diagonal elements of the diagonal matrix D from the factorization A = LDL^H.
[in,out]	E()	Array E(LE - 1) (LE >= N - 1) [in] N-1 sub-diagonal elements of the Hermintian positive definite tridiagonal matrix A. [out] N-1 sub-diagonal elements of the unit bidiagonal factor L from the LDL^H factorization of A. E can also be regarded as the super-diagonal of the unit bidiagonal factor U from the U^HDU factorization of A.
[out]	Info	= 0: Successful exit. = -1: The argument N had an illegal value. (N < 0) = -2: The argument D() is invalid. = -3: The argument E() is invalid. = i > 0: The leading minor of order i is not positive definite. If i < N, the factorization could not been completed, while if i = N, the factorization was completed, but D(N-1) <= 0.

Reference: LAPACK

Example Program: Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where
( 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 )

Sub Ex_Zpttrf()

Const N As Long = 3

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

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

D(0) = 2.88: D(1) = 0.62: D(2) = 0.46

E(0) = Cmplx(0.29, 0.44): E(1) = Cmplx(-0.01, 0.02)

B(0) = Cmplx(1.6236, -0.73): B(1) = Cmplx(0.1581, 0.1537): B(2) = Cmplx(0.1132, -0.229)

ANorm = Zlanht("1", N, D(), E())

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

If Info = 0 Then Call Zpttrs("L", N, D(), E(), B(), Info)

If Info = 0 Then Call Zptcon(N, D(), E(), 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

Cmplx
Function Cmplx(R As Double, Optional I As Double=0) As Complex
Building complex number

Cimag
Function Cimag(A As Complex) As Double
Imaginary part of complex number

Creal
Function Creal(A As Complex) As Double
Real part of complex number

Zlanht
Function Zlanht(Norm As String, N As Long, D() As Double, E() As Complex, Optional Info As Long) As Double
One norm, Frobenius norm, infinity norm, or largest absolute value of any element of a Hermitian trid...

Zptcon
Sub Zptcon(N As Long, D() As Double, E() As Complex, ANorm As Double, RCond As Double, Info As Long)
Condition number of a Hermitian positive definite tridiagonal matrix

Zpttrf
Sub Zpttrf(N As Long, D() As Double, E() As Complex, Info As Long)
LDLH factorization of a Hermitian positive definite tridiagonal matrix

Zpttrs
Sub Zpttrs(Uplo As String, N As Long, D() As Double, E() As Complex, B() As Complex, Info As Long, Optional Nrhs As Long=1)
Solution to factorized system of linear equations AX = B for a Hermitian positive definite tridiagona...

Example Results: X = 0.59 -0.28 -0.2 -0.04 0.24 -0.49

RCond = 0.124521368143895

Info = 0