Zhecon Zhetrf Zhetri Zhetrs Zhpcon Zhptrf Zhptri Zhptrs

Zhetrf() Sub Zhetrf ( Uplo As  String, N As  Long, A() As  Complex, IPiv() As  Long, Info As  Long  ) UDUH or LDLH factorization of a Hermitian matrix Purpose This routine computes the factorization of a Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U^H or A = L*D*L^H where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1 x 1 and 2 x 2 diagonal blocks. 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 matrix A. The upper or lower triangular part is to be referenced in accordance with Uplo. [out] The block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details). [out] IPiv() Array IPiv(LIPiv - 1) (LIPiv >= N) Details of the interchanges and the block structure of D. If IPiv(k-1) > 0, then rows and columns k and IPiv(k-1) were interchanged, and k-th diagonal of D is a 1 x 1 diagonal block. If Uplo = "U" and IPiv(k-1) = IPiv(k-2) < 0, then rows and columns k-1 and -IPiv(k-1) were interchanged and (k-1)-th diagonal of D is a 2 x 2 diagonal block. If Uplo = "L" and IPiv(k-1) = IPiv(k) < 0, then rows and columns k+1 and -IPiv(k-1) were interchanged and k-th diagonal of D is a 2 x 2 diagonal block. [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. = -4: The argument IPiv() is invalid. = i > 0: The i-th diagonal element of D is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations. Further Details If Uplo = "U", then A = U*D*U^H, where U = P(n)*U(n)* ... *P(k)U(k)* ..., i.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1 x 1 and 2 x 2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPiv(k-1), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I v 0 ) k-s U(k) = ( 0 I 0 ) s ( 0 0 I ) n-k k-s s n-k If s = 1, D(k) overwrites A(k-1, k-1), and v overwrites A(0 to k-2, k-1). If s = 2, the upper triangle of D(k) overwrites A(k-2, k-2), A(k-2, k-1), and A(k-1, k-1), and v overwrites A(0 to k-3, k-2 to k-1). If Uplo = "L", then A = L*D*L^H, where L = P(1)*L(1)* ... *P(k)*L(k)* ..., i.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1 x 1 and 2 x 2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPiv(k-1), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then ( I 0 0 ) k-1 L(k) = ( 0 I 0 ) s ( 0 v I ) n-k-s+1 k-1 s n-k-s+1 If s = 1, D(k) overwrites A(k-1, k-1), and v overwrites A(k to n-1, k-1). If s = 2, the lower triangle of D(k) overwrites A(k-1, k-1), A(k-1, k), and A(k, k), and v overwrites A(k+1 to n-1, k-1 to k). Reference LAPACK Example Program Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where ( 0.20 -0.11+0.93i 0.81-0.37i ) A = ( -0.11-0.93i -0.32 -0.80+0.92i ) ( 0.81+0.37i -0.80-0.92i -0.29 ) ( -0.1220+0.1844i ) B = ( 0.0034-0.4346i ) ( 0.5339-0.1571i ) Sub Ex_Zhetrf() Const N As Long = 3 Dim A(N - 1, N - 1) As Complex, B(N - 1) As Complex, IPiv(N - 1) As Long Dim ANorm As Double, RCond As Double, Info As Long A(0, 0) = Cmplx(0.2, 0) A(1, 0) = Cmplx(-0.11, -0.93): A(1, 1) = Cmplx(-0.32, 0) A(2, 0) = Cmplx(0.81, 0.37): A(2, 1) = Cmplx(-0.8, -0.92): A(2, 2) = Cmplx(-0.29, 0) B(0) = Cmplx(-0.122, 0.1844): B(1) = Cmplx(0.0034, -0.4346): B(2) = Cmplx(0.5339, -0.1571) ANorm = Zlanhe("1", "L", N, A()) Call Zhetrf("L", N, A(), IPiv(), Info) If Info = 0 Then Call Zhetrs("L", N, A(), IPiv(), B(), Info) If Info = 0 Then Call Zhecon("L", N, A(), IPiv(), 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 Example Results X = 0.86 0.64 0.51 0.71 0.590000000000001 -0.15 RCond = 4.46158691608911E-02 Info = 0