◆ Zpotrf()

Sub Zpotrf	(	Uplo As	String,
		N As	Long,
		A() As	Complex,
		Info As	Long
	)

Cholesky factorization of a Hermitian positive definite matrix

Purpose: This routine computes the Cholesky factorization of a Hermitian positive definite matrix A. 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.
This is the blocked version of the algorithm, calling Level 3 BLAS.

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] N x N Hermitian positive definite matrix A. The upper or lower triangular part is to be referenced 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 A() 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_Zpotrf()

Const N As Long = 3

Dim A(N - 1, N - 1) As Complex, B(N - 1) As Complex

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

A(0, 0) = Cmplx(2.2, 0)

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

A(2, 0) = Cmplx(0.81, 0.37): A(2, 1) = Cmplx(-0.8, -0.92): A(2, 2) = 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 = Zlanhe("1", "L", N, A())

Call Zpotrf("L", N, A(), Info)

If Info = 0 Then Call Zpotrs("L", N, A(), B(), Info)

If Info = 0 Then Call Zpocon("L", N, A(), 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

Zlanhe
Function Zlanhe(Norm As String, Uplo As String, N As Long, A() As Complex, Optional Info As Long) As Double
One norm, Frobenius norm, infinity norm, or largest absolute value of any element of a Hermitian matr...

Zpotrf
Sub Zpotrf(Uplo As String, N As Long, A() As Complex, Info As Long)
Cholesky factorization of a Hermitian positive definite matrix

Zpotrs
Sub Zpotrs(Uplo As String, N As Long, A() 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 matrix

Zpocon
Sub Zpocon(Uplo As String, N As Long, A() As Complex, ANorm As Double, RCond As Double, Info As Long)
Condition number of a Hermitian positive definite matrix

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

RCond = 8.85790434328042E-02

Info = 0