◆ Zgeesx()

Sub Zgeesx	(	JobVs As	String,
		Sort As	String,
		Selct As	LongPtr,
		Sense As	String,
		N As	Long,
		A() As	Complex,
		Sdim As	Long,
		W() As	Complex,
		Vs() As	Complex,
		RConde As	Double,
		RCondv As	Double,
		Info As	Long
	)

(Expert driver) Schur factorization of a complex matrix

Purpose: This routine computes for an n x n complex nonsymmetric matrix A, the eigenvalues, the Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*Z^H.

Optionally, it also orders the eigenvalues on the diagonal of the Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of Z form an orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.8.1 of the LAPACK Users' Guide Third Edition (where these quantities are called s and sep respectively).

A complex matrix is in Schur form if it is upper triangular.

Parameters

[in]	JobVs	= "N": Schur vectors are not computed = "V": Schur vectors are computed
[in]	Sort	Specifies whether or not to order the eigenvalues on the diagonal of the Schur form = "N": Eigenvalues are not ordered = "S": Eigenvalues are ordered (see Selct)
[in]	Selct	Sort = "S": Selct is used to select eigenvalues to sort to the top left of the Schur form. An eigenvalue W(j) is selected if Selct(W(j)) is true (= 1). Sort = "N": Selct is not referenced.
[in]	Sense	Determines which reciprocal condition numbers are computed. = 'N': None are computed. = 'E': Computed for average of selected eigenvalues only. = 'V': Computed for selected right invariant subspace only. = 'B': Computed for both. If Sense = "E", "V" or "B", Sort must equal "S"
[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 matrix A. [out] A() has been overwritten by its Schur form T.
[out]	Sdim	Sort = "N": Sdim = 0. Sort = "S": Sdim = number of eigenvalues for which Selct is true.
[out]	W()	Array W(LW - 1) (LW >= N) W() contains the computed eigenvalues, in the same order that they appear on the diagonal of the output Schur form T.
[out]	Vs()	Array Vs(LVs1 - 1, LVs2 - 1) (LVs1 >= N, LVs2 >= N) Jobvs = "V": Vs() contains the unitary matrix Z of Schur vectors. Jobvs = "N": Vs() is not referenced.
[out]	RConde	Sense = 'E' or 'B': RConde contains the reciprocal condition number for the average of the selected eigenvalues. Sense = 'N' or 'V': Not referenced.
[out]	RCondv	Sense = 'V' or 'B': RCondv contains the reciprocal condition number for the selected right invariant subspace. Sense = 'N' or 'E': Not referenced.
[out]	Info	= 0: Successful exit. = -1: The argument Jobvs had an illegal value. (Jobvs <> "V" nor "N") = -2: The argument Sort had an illegal value. (Sort <> "S" nor "N") = -4: The argument Sense had an illegal value. (Sense <> "E", "V", "B" nor "N") = -5: The argument N had an illegal value. (N < 0) = -6: The argument A() is invalid. = -8: The argument W() is invalid. = -9: The argument Vs() is invalid. = i (0 < i <= N): The QR algorithm failed to compute all the eigenvalues. Elements 0 to Ilo-2 and i to N-1 of W() contain those eigenvalues which have converged. If Jobvs = "V", Vs() contains the matrix which reduces A to its partially converged Schur form. = N+1: The eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is very ill-conditioned). = N+2: After reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Schur form no longer satisfy Selct = true. This could also be caused by underflow due to scaling.

Reference: LAPACK

Example Program: Compute all eigenvalues, Schur form T, and, Schur vectors of the general matrix A, where
( 0.20-0.11i -0.93-0.32i 0.81+0.37i )

A = ( -0.80-0.92i -0.29+0.86i 0.64+0.51i )

( 0.71+0.59i -0.15+0.19i 0.20+0.94i )

Sub Ex_Zgeesx()

Const N = 3

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

Dim Sdim As Long, Vs(N - 1, N - 1) As Complex

Dim RConde As Double, RCondv As Double, Info As Long

A(0, 0) = Cmplx(0.2, -0.11): A(0, 1) = Cmplx(-0.93, -0.32): A(0, 2) = Cmplx(0.81, 0.37)

A(1, 0) = Cmplx(-0.8, -0.92): A(1, 1) = Cmplx(-0.29, 0.86): A(1, 2) = Cmplx(0.64, 0.51)

A(2, 0) = Cmplx(0.71, 0.59): A(2, 1) = Cmplx(-0.15, 0.19): A(2, 2) = Cmplx(0.2, 0.94)

Call Zgeesx("V", "S", AddressOf Selct, "B", N, A(), Sdim, W(), Vs(), RConde, RCondv, Info)

Debug.Print "Eigenvalues ="

Debug.Print Creal(W(0)), Cimag(W(0)), Creal(W(1)), Cimag(W(1))

Debug.Print Creal(W(2)), Cimag(W(2))

Debug.Print "Schur form T ="

Debug.Print Creal(A(0, 0)), Cimag(A(0, 0)), Creal(A(0, 1)), Cimag(A(0, 1))

Debug.Print Creal(A(1, 0)), Cimag(A(1, 0)), Creal(A(1, 1)), Cimag(A(1, 1))

Debug.Print Creal(A(2, 0)), Cimag(A(2, 0)), Creal(A(2, 1)), Cimag(A(2, 1))

Debug.Print Creal(A(0, 2)), Cimag(A(0, 2))

Debug.Print Creal(A(1, 2)), Cimag(A(1, 2))

Debug.Print Creal(A(2, 2)), Cimag(A(2, 2))

Debug.Print "Schur vectors ="

Debug.Print Creal(Vs(0, 0)), Cimag(Vs(0, 0)), Creal(Vs(0, 1)), Cimag(Vs(0, 1))

Debug.Print Creal(Vs(1, 0)), Cimag(Vs(1, 0)), Creal(Vs(1, 1)), Cimag(Vs(1, 1))

Debug.Print Creal(Vs(2, 0)), Cimag(Vs(2, 0)), Creal(Vs(2, 1)), Cimag(Vs(2, 1))

Debug.Print Creal(Vs(0, 2)), Cimag(Vs(0, 2))

Debug.Print Creal(Vs(1, 2)), Cimag(Vs(1, 2))

Debug.Print Creal(Vs(2, 2)), Cimag(Vs(2, 2))

Debug.Print "Rconde =", RConde, "Rcondv =", RCondv

Debug.Print "Sdim =", Sdim, "Info =", Info

End Sub

Function Selct(W As Complex) As Long

Selct = 0

If Cimag(W) <> 0 Then Selct = 1

End Function

Example Results: Eigenvalues =

-1.15894122423918 -0.50662892448174 1.05593587167591 0.900255855387815

0.21300535256327 1.29637306909393

Schur form T =

-1.15894122423918 -0.50662892448174 2.31358655627147E-02 -0.442808752291521

0 0 1.05593587167591 0.900255855387815

0 0 0 0

-0.644484909898823 -0.733094546116673

0.379301413619788 -0.286520422490734

0.21300535256327 1.29637306909393

Schur vectors =

-0.474826946245658 0.464530398931381 -0.103329315153953 0.628092734605153

-0.39444507925059 0.539925475846714 -9.30826818180923E-02 -0.290418461635993

0.264837176182039 -0.203729501266495 -0.367753983153771 0.605452152301986

1.03227229675376E-02 0.391748503946406

2.12814186266949E-02 -0.67781516115713

0.349266221164391 -0.514347515136026

Rconde = 1 Rcondv = 2.76522085257846

Sdim = 3 Info = 0