◆ Dpotrf()

Sub Dpotrf	(	Uplo As	String,
		N As	Long,
		A() As	Double,
		Info As	Long
	)

Cholesky factorization of a symmetric positive definite matrix

Purpose: This routine computes the Cholesky factorization of a real symmetric positive definite matrix A. The factorization has the form
A = U^T * U, if uplo = "U", or

A = L * L^T, 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 symmetric 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^TU or A = LL^T.
[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 A is symmetric positive definite and
( 2.2 -0.11 -0.32 ) ( -1.566 )

A = ( -0.11 2.93 0.81 ), B = ( -2.8425 )

( -0.32 0.81 2.37 ) ( -1.1765 )

Sub Ex_Dpotrf()

Const N = 3

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

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

A(0, 0) = 2.2

A(1, 0) = -0.11: A(1, 1) = 2.93

A(2, 0) = -0.32: A(2, 1) = 0.81: A(2, 2) = 2.37:

B(0) = -1.566: B(1) = -2.8425: B(2) = -1.1765

ANorm = Dlansy("1", "L", N, A())

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

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

If Info = 0 Then Call Dpocon("L", N, A(), ANorm, RCond, Info)

Debug.Print "X =", B(0), B(1), B(2)

Debug.Print "RCond =", RCond

Debug.Print "Info =", Info

End Sub

Dlansy
Function Dlansy(Norm As String, Byval Uplo As String, N As Long, A() As Double, Optional Info As Long) As Double
One norm, Frobenius norm, infinity norm, or largest absolute value of any element of a real symmetric...

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

Dpotrs
Sub Dpotrs(Uplo As String, N As Long, A() As Double, B() As Double, Info As Long, Optional Nrhs As Long=1)
Solution to factorized system of linear equations AX = B for a symmetric positive definite matrix

Dpotrf
Sub Dpotrf(Uplo As String, N As Long, A() As Double, Info As Long)
Cholesky factorization of a symmetric positive definite matrix

Example Results: X = -0.8 -0.92 -0.29

RCond = 0.446791078068956

Info = 0