◆ Zpttrf()

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

係数行列のLDL^H分解 (正定値エルミート3重対角行列)

目的: 本ルーチンは正定値エルミート3重対角行列AのL*D*L^H分解を計算する. この分解はまた A = U^H*D*U の形式とみなすこともできる.

引数

[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]	D()	配列 D(LD - 1) (LD >= N) [in] 正定値エルミート3重対角行列AのN個の対角要素. [out] 分解 A = LDL^H の対角行列DのN個の対角要素.
[in,out]	E()	配列 E(LE - 1) (LE >= N - 1) [in] 正定値エルミート3重対角行列AのN-1個の副対角要素. [out] 分解 A = LDL^H の下2重対角行列LのN-1個の下副対角要素. これは, 分解 A = U^HDU の上2重対角行列Uの上副対角要素とみなすこともできる(LおよびUの対角要素は1).
[out]	Info	= 0: 正常終了. = -1: パラメータ N の誤り. (N < 0) = -2: パラメータ D() の誤り. = -3: パラメータ E() の誤り. = i > 0: i×i首座小行列が正定値でない. i < N の場合, 分解を完了できなかった. i = N の場合, 分解は完了したが D(N-1) <= 0 である.

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,
( 2.88 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 )

とする.
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
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

Zlanht
Function Zlanht(Norm As String, N As Long, D() As Double, E() 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 (エルミート3重対角行列)

Zptcon
Sub Zptcon(N As Long, D() As Double, E() 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3重対角行列)

Zpttrf
Sub Zpttrf(N As Long, D() As Double, E() As Complex, Info As Long)
係数行列のLDLH分解 (正定値エルミート3重対角行列)

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)
分解済の連立一次方程式 AX = B の解 (正定値エルミート3重対角行列)

実行結果: X = 0.59 -0.28 -0.2 -0.04 0.24 -0.49

RCond = 0.124521368143895

Info = 0