Zcgesv Zgbsv Zgbsvx Zgesv Zgesvx Zgtsv Zgtsvx Zspsv Zspsvx Zsysv Zsysvx

Zgtsvx() Sub Zgtsvx ( Fact As  String, Trans As  String, N As  Long, Dl() As  Complex, D() As  Complex, Du() As  Complex, Dlf() As  Complex, Df() As  Complex, Duf() As  Complex, Du2() As  Complex, IPiv() As  Long, B() As  Complex, X() As  Complex, RCond As  Double, FErr() As  Double, BErr() As  Double, Info As  Long, Optional Nrhs As  Long = 1  ) (Expert driver) Solution to system of linear equations AX = B for a complex tridiagonal matrix Purpose This routine uses the LU factorization to computes the solution to a complex system of linear equations A * X = B or A^T * X = B where A is a tridiagonal matrix of order n and X and B are n x nrhs matrices. Error bounds on the solution and a condition estimate are also provided. Description The following steps are performed: If Fact = "N", the LU decomposition is used to factor the matrix A as A = L * U where L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with nonzeros in only the main diagonal and first two super-diagonals. If i-th diagonal element of U = 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = n+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. The system of equations is solved for X using the factored form of A. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. Parameters [in] Fact Specifies whether or not the factored form of A has been supplied on entry. = "F": Dlf(), Df(), Duf(), Du2() and IPiv() contain the factored form of A; Dlf(), Df(), Duf(), Du2() and IPiv() will not be modified. = "N": The matrix will be copied to Dlf(), Df() and Duf() and factored. [in] Trans Specifies the form of the system of equations: = "N": A * X = B. (no transpose) = "T" or "C": A^T * X = B. (transpose) [in] N Order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in] Dl() Array Dl(LDl - 1) (LDl >= N - 1) N-1 sub-diagonal elements of A. [in] D() Array D(LD - 1) (LD >= N) N diagonal elements of A. [in] Du() Array Du(LDu - 1) (LDu >= N - 1) N-1 super-diagonal elements of A. [in,out] Dlf() Array Dlf(LDlf - 1) (LDlf >= N - 1) [in] If Fact = "F", N-1 multipliers that define the matrix L from the LU factorization of A as computed by Dgttrf, are to be stored. [out] If Fact = "N", N-1 multipliers that define the matrix L from the LU factorization of A, are returned. [in,out] Df() Array Df(LDf - 1) (LDf >= N) [in] If Fact = "F", diagonal elements of the upper triangular matrix U from the LU factorization of A, are to be stored. [out] If Fact = "N", diagonal elements of the upper triangular matrix U from the LU factorization of A, are returned. [in,out] Duf() Array Duf(LDuf - 1) (LDuf >= N - 1) [in] If Fact = "F", N-1 elements of the first super-diagonal of U, are to be stored. [out] If Fact = "N", N-1 elements of the first super-diagonal of U, are returned. [in,out] Du2() Array Du2(LDu2 - 1) (LDu2 >= N - 2) [in] If Fact = "F", N-2 elements of the second super-diagonal of U, are to be stored. [out] If Fact = "N", N-2 elements of the second super-diagonal of U, are returned. [in,out] IPiv() Array IPiv(LIPiv - 1) (LIPiv >= N) [in] If Fact = "F", the pivot indices from the LU factorization of A as computed by Dgttrf, are to be stored. [out] If Fact = "N", the pivot indices from the LU factorization of A are returned; row i of the matrix was interchanged with row IPiv(i-1). IPiv(i-1) will always be either i or i+1; IPiv(i-1) = i indicates a row interchange was not required. [in] B() Array B(LB1 - 1, LB2 - 1) (LB1 >= max(1, N), LB2 >= Nrhs) (2D array) or B(LB - 1) (LB >= max(1, N), Nrhs = 1) (1D array) N x Nrhs matrix of right hand side matrix B. [out] X() Array X(LX1 - 1, LX2 - 1) (LX1 >= max(1, N), LX2 >= Nrhs) (2D array) or X(LX - 1) (LX >= max(1, N), Nrhs = 1) (1D array) If Info = 0 or Info = N+1, the N x Nrhs solution matrix X. [out] RCond The estimate of the reciprocal condition number of the matrix A. If RCond is less than the machine precision (in particular, if RCond = 0), the matrix is singular to working precision. This condition is indicated by a return code of Info > 0. [out] FErr() Array FErr(LFErr - 1) (LFErr >= Nrhs) The estimated forward error bound for each solution vector X(j) (the j-th column of the solution matrix X). If Xtrue is the true solution corresponding to X(j), FErr(j-1) is an estimated upper bound for the magnitude of the largest element in (X(j) - Xtrue) divided by the magnitude of the largest element in X(j). The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error. [out] BErr() Array BErr(LBErr - 1) (LBErr >= Nrhs) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any element of A or B that makes X(j) an exact solution). [out] Info = 0: Successful exit. = -1: The argument Fact had an illegal value. (Fact <> "F" nor "N") = -2: The argument Trans had an illegal value. (Trans <> "N", "T" nor "C") = -3: The argument N had an illegal value. (N < 0) = -4: The argument Dl() is invalid. = -5: The argument D() is invalid. = -6: The argument Du() is invalid. = -7: The argument Dlf() is invalid. = -8: The argument Df() is invalid. = -9: The argument Duf() is invalid. = -10: The argument Du2() is invalid. = -11: The argument IPiv() is invalid. = -12: The argument B() is invalid. = -13: The argument X() is invalid. = -15: The argument FErr() is invalid. = -16: The argument BErr() is invalid. = -18: The argument Nrhs had an illegal value. (Nrhs < 0) = i (0 < i <= N): The i-th diagonal element of the factor U is exactly zero. The factorization has not been completed unless i = N, but the factor U is exactly singular, so the solution and error bounds could not be computed. RCond = 0 is returned. = N+1: U is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCond would suggest. [in] Nrhs (Optional) Number of right hand sides, i.e., number of columns of the matrix B. (Nrhs >= 0) (If Nrhs = 0, returns without computation) (default = 1) Reference LAPACK Example Program Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where ( 0.81+0.37i -0.20-0.11i 0 ) A = ( 0.64+0.51i -0.80-0.92i -0.93-0.32i ) ( 0 0.71+0.59i -0.29+0.86i ) ( -0.0484+0.2644i ) B = ( -0.2644-1.0228i ) ( -0.5299+1.5025i ) Sub Ex_Zgtsvx() Const N = 3 Dim Dl(N - 2) As Complex, D(N - 1) As Complex, Du(N - 2) As Complex Dim Dlf(N - 2) As Complex, Df(N - 1) As Complex, Duf(N - 2) As Complex Dim Du2(N - 3) As Complex, IPiv(N - 1) As Long Dim B(N - 1) As Complex, X(N - 1) As Complex Dim FErr(0) As Double, BErr(0) As Double Dim RCond As Double, Info As Long Dl(0) = Cmplx(0.64, 0.51): Dl(1) = Cmplx(0.71, 0.59) D(0) = Cmplx(0.81, 0.37): D(1) = Cmplx(-0.8, -0.92): D(2) = Cmplx(-0.29, 0.86) Du(0) = Cmplx(0.2, -0.11): Du(1) = Cmplx(-0.93, -0.32) B(0) = Cmplx(-0.0484, 0.2644): B(1) = Cmplx(-0.2644, -1.0228): B(2) = Cmplx(-0.5299, 1.5025) Call Zgtsvx("N", "N", N, Dl(), D(), Du(), Dlf(), Df(), Duf(), Du2(), IPiv(), B(), X(), RCond, FErr(), BErr(), Info) Debug.Print "X =", Debug.Print Creal(X(0)), Cimag(X(0)), Creal(X(1)), Cimag(X(1)), Creal(X(2)), Cimag(X(2)) Debug.Print "RCond =", RCond Debug.Print "Info =", Info End Sub Example Results X = -0.15 0.19 0.2 0.94 0.79 -0.13 RCond = 0.187722560135325 Info = 0