◆ Zpptrf()

Sub Zpptrf	(	Uplo As	String,
		N As	Long,
		Ap() As	Complex,
		Info As	Long
	)

Cholesky factorization of a Hermitian positive definite matrix in packed form

Purpose: This routine computes the Cholesky factorization of a Hermitian positive definite matrix A stored in packed form. The factorization has the form
A = U^H * U, if uplo = "U", or

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

where U is an upper triangular matrix and L is a lower triangular matrix.

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]	Ap()	Array Ap(LAp - 1) (LAp >= N(N + 1)/2) [in] N x N Hermitian positive definite matrix A in packed form. The upper or lower part is to be stored in accordance with Uplo. [out] If Info = 0, the factor U or L from the Cholesky factorization A = U^HU or A = LL^H.
[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 Ap() is invalid. = i > 0: The leading minor of order i is not positive definite, and the factorization could not be completed.

Reference: LAPACK

Example Program: Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where
( 2.20 -0.11+0.93i 0.81-0.37i )

A = ( -0.11-0.93i 2.32 -0.80+0.92i )

( 0.81+0.37i -0.80-0.92i 2.29 )

( 1.5980+1.4644i )

B = ( 1.3498+1.4398i )

( 2.0561-0.5441i )

Sub Ex_Zpptrf()

Const N As Long = 3

Dim Ap(N * (N + 1) / 2) As Complex, B(N - 1) As Complex

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

Ap(0) = Cmplx(2.2, 0)

Ap(1) = Cmplx(-0.11, -0.93): Ap(3) = Cmplx(2.32, 0)

Ap(2) = Cmplx(0.81, 0.37): Ap(4) = Cmplx(-0.8, -0.92): Ap(5) = Cmplx(2.29, 0)

B(0) = Cmplx(1.598, 1.4644): B(1) = Cmplx(1.3498, 1.4398): B(2) = Cmplx(2.0561, -0.5441)

ANorm = Zlansp("1", "L", N, Ap())

Call Zpptrf("L", N, Ap(), Info)

If Info = 0 Then Call Zpptrs("L", N, Ap(), B(), Info)

If Info = 0 Then Call Zppcon("L", N, Ap(), 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

Zlansp
Function Zlansp(Norm As String, Uplo As String, N As Long, Ap() As Complex, Optional Info As Long) As Double
One norm, Frobenius norm, infinity norm, or largest absolute value of any element of a complex symmet...

Zpptrf
Sub Zpptrf(Uplo As String, N As Long, Ap() As Complex, Info As Long)
Cholesky factorization of a Hermitian positive definite matrix in packed form

Zppcon
Sub Zppcon(Uplo As String, N As Long, Ap() As Complex, ANorm As Double, RCond As Double, Info As Long)
Condition number of a Hermitian positive definite matrix in packed form

Zpptrs
Sub Zpptrs(Uplo As String, N As Long, Ap() 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 matrix in packed form

Example Results: X = 0.86 0.64 0.51 0.71 0.59 -0.15

RCond = 8.85790434328042E-02

Info = 0