◆ Zpptrf()

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

係数行列のコレスキー分解 (正定値エルミート行列) (圧縮形式)

目的: 本ルーチンは圧縮形式の正定値エルミート行列Aのコレスキー分解を計算する. 分解は次の形式である.
A = U^H * U (Uplo = "U"の場合)

A = L * L^H (Uplo = "L"の場合)

ここで, Uは上三角行列, Lは下三角行列である.

引数

[in]	Uplo	= "U": Aの上三角部分を格納. = "L": Aの下三角部分を格納.
[in]	N	行列Aの行および列数. (N >= 0) (N = 0 の場合, 処理を行わずに戻る)
[in,out]	Ap()	配列 Ap(LAp - 1) (LAp >= N(N + 1)/2) [in] 圧縮形式のN×N正定値エルミート行列 A. Uploに従い上三角部分または下三角部分を格納する. [out] Info = 0 の場合, コレスキー分解 A = U^HU または A = LL^H の UまたはL.
[out]	Info	= 0: 正常終了. = -1: パラメータ Uplo の誤り. (Uplo <> "U"および"L") = -2: パラメータ N の誤り. (N < 0) = -3: パラメータ Ap() の誤り. = i > 0: i×i首座小行列が正定値でないため分解を完了できなかった.

出典: LAPACK

使用例: 連立一次方程式 Ax = B を解き, 同時にAの条件数の逆数の推定値(RCond)を求める. ただし,
( 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 )

とする.
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
複素数の作成

Cimag
Function Cimag(A As Complex) As Double
複素数の虚数部

Creal
Function Creal(A As Complex) As Double
複素数の実数部

Zlansp
Function Zlansp(Norm As String, Uplo As String, N As Long, Ap() As Complex, Optional Info As Long) As Double
行列の1ノルム, フロベニウスノルム, 無限ノルム, または, 要素の最大絶対値 (複素対称行列) (圧縮形式)

Zpptrf
Sub Zpptrf(Uplo As String, N As Long, Ap() As Complex, Info As Long)
係数行列のコレスキー分解 (正定値エルミート行列) (圧縮形式)

Zppcon
Sub Zppcon(Uplo As String, N As Long, Ap() As Complex, ANorm As Double, RCond As Double, Info As Long)
行列の条件数 (正定値エルミート行列) (圧縮形式)

Zpptrs
Sub Zpptrs(Uplo As String, N As Long, Ap() As Complex, B() As Complex, Info As Long, Optional Nrhs As Long=1)
分解済の連立一次方程式 AX = B の解 (正定値エルミート行列) (圧縮形式)

実行結果: X = 0.86 0.64 0.51 0.71 0.59 -0.15

RCond = 8.85790434328042E-02

Info = 0