Dpbsv Dpbsvx Dptsv Dptsvx

Dptsvx() Sub Dptsvx ( Fact As  String, N As  Long, D() As  Double, E() As  Double, Df() As  Double, Ef() As  Double, B() As  Double, X() As  Double, 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 symmetric positive definite tridiagonal matrix Purpose This routine uses the factorization A = L*D*L^T to compute the solution to a real system of linear equations A * X = B, where A is an n x n symmetric positive definite tridiagonal matrix, 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 matrix A is factored as A = L * D * L^T, where L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as having the form A = U^T * D * U. If the leading i x i principal minor is not positive definite, 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": Df() and Ef() contain the factored form of A. Df() and Ef() will not be modified. = "N": The matrix A will be copied to Df() and Ef() and factored. [in] N Order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in] D() Array D(LD - 1) (LD >= N) N diagonal elements of the symmetric positive definite tridiagonal matrix A. [in] E() Array E(LE - 1) (LE >= N - 1) N-1 sub-diagonal elements of the symmetric positive definite tridiagonal matrix A. [in,out] Df() Array Df(LDf - 1) (LDf >= N) [in] If Fact = "F", n diagonal elements of the diagonal matrix D from the L*D*L^T factorization of A, are to be stored. [out] If Fact = "N", n diagonal elements of the diagonal matrix D from the L*D*L^T factorization of A, are returned. [in,out] Ef() Array Ef(LEf - 1) (LEf >= N - 1) [in] If Fact = "F", N-1 sub-diagonal elements of the unit bidiagonal factor L from the L*D*L^T factorization of A, are to be stored. [out] If Fact = "N", N-1 sub-diagonal elements of the unit bidiagonal factor L from the L*D*L^T factorization of A, are returned. [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 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 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 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). [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 N had an illegal value. (N < 0) = -3: The argument D() is invalid. = -4: The argument E() is invalid. = -5: The argument Df() is invalid. = -6: The argument Ef() is invalid. = -7: The argument B() is invalid. = -8: The argument X() is invalid. = -10: The argument FErr() is invalid. = -11: The argument BErr() is invalid. = -13: The argument Nrhs had an illegal value. (Nrhs < 0) [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 A is symmetric positive definite tridiagonal matrix and ( 2.58 -0.99 0 ) ( -1.1850 ) A = ( -0.99 0.69 -0.03 ), B = ( 0.1410 ) ( 0 -0.03 0.18 ) ( 0.1614 ) Sub Ex_Dptsvx() Const N As Long = 3 Dim D(N - 1) As Double, E(N - 2) As Double, B(N - 1) As Double Dim Df(N - 1) As Double, Ef(N - 2) As Double, X(N - 1) As Double Dim FErr(0) As Double, BErr(0) As Double Dim RCond As Double, Info As Long D(0) = 2.58: D(1) = 0.69: D(2) = 0.18 E(0) = -0.99: E(1) = -0.03 B(0) = -1.185: B(1) = 0.141: B(2) = 0.1614 Call Dptsvx("N", N, D(), E(), Df(), Ef(), B(), X(), RCond, FErr(), BErr(), Info) Debug.Print "X =", X(0), X(1), X(2) Debug.Print "RCond =", RCond Debug.Print "Info =", Info End Sub Example Results X = -0.82 -0.94 0.74 RCond = 0.0437508336668 Info = 0