Dspcon Dsptrf Dsptri Dsptrs Dsycon Dsytrf Dsytri Dsytrs

Dsytrf() Sub Dsytrf ( Uplo As  String, N As  Long, A() As  Double, IPiv() As  Long, Info As  Long  ) UDUT or LDLT factorization of a symmetric matrix Purpose This routine computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is A = U*D*U^T or A = L*D*L^T 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 symmetric 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^T, 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^T, 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 ( 2.2 -0.11 -0.32 ) ( -1.5660 ) A = ( -0.11 2.93 0.81 ), B = ( -2.8425 ) ( -0.32 0.81 -2.37 ) ( -1.1765 ) Sub Ex_Dsytrf() Const N As Long = 3 Dim A(N - 1, N - 1) As Double, B(N - 1) As Double, IPiv(N - 1) As Long 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 Dsytrf("L", N, A(), IPiv(), Info) If Info = 0 Then Call Dsytrs("L", N, A(), IPiv(), B(), Info) If Info = 0 Then Call Dsycon("L", N, A(), IPiv(), ANorm, RCond, Info) Debug.Print "X =", B(0), B(1), B(2) Debug.Print "RCond =", RCond Debug.Print "Info =", Info End Sub Example Results X = -0.8 -0.92 -0.29 RCond = 0.446791078068956 Info = 0