Zcposv Zposvx Zppsv Zppsvx

Zcposv() Sub Zcposv ( Uplo As  String, N As  Long, A() As  Complex, B() As  Complex, X() As  Complex, Iter As  Long, Info As  Long, Optional Nrhs As  Long = 1  ) (Simple driver) Solution to system of linear equations AX = B for a Hermitian positive definite matrix (mixed precision with iterative refinement) Purpose This routine computes the solution to a complex system of linear equations A * X = B, where A is an n x n Hermitian positive definite matrix and X and B are n x nrhs matrices. Zcposv first attempts to factorize the matrix in single precision and use this factorization within an iterative refinement procedure to produce a solution with double precision normwise backward error quality (see below). If the approach fails the method switches to a double precision factorization and solve. The iterative refinement is not going to be a winning strategy if the ratio single precision performance over double precision performance is too small. A reasonable strategy should take the number of right-hand sides and the size of the matrix into account. Up to now, we always try iterative refinement. The iterative refinement process is stopped if   iter > itermax or for all the rhs we have:   rnrm < sqrt(n)*xnrm*anrm*eps*bwdmax where   iter is the number of the current iteration in the iterative refinement process   rnrm is the infinity-norm of the residual   xnrm is the infinity-norm of the solution   anrm is the infinity-operator-norm of the matrix A   eps is the machine epsilon returned by dlamch('E') The value itermax and bwdmax are fixed to 30 and 1.0 respectively. Parameters [in] Uplo = "U": Upper triangle of A is stored. = "L": Lower triangle of A is stored. [in] N Number of linear equations, i.e., 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 positove definite matrix A. The upper or lower triangular part is to be referenced in accordance with Uplo. [out] If iterative refinement has been successfully used (Info = 0 and Iter >= 0, see description below), then A() is unchanged. If double precision factorization has been used (Info = 0 and Iter < 0, see description below), then the array A() contains the factor U or L from the Cholesky factorization A = U^H*U or A = L*L^H. [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, the N x Nrhs solution matrix X. [out] Iter < 0: Iterative refinement has failed, double precision factorization has been performed.   = -1: The routine fell back to full precision for implementation- or machine-specific reasons.   = -2: Narrowing the precision induced an overflow, the routine fell back to full precision.   = -3: Failure of Sgetrf.   = -31: Stop the iterative refinement after the 30th iterations. > 0: Iterative refinement has been successfully used. Returns the number of iterations. [out] Info = 0: Successful exit. = -2: The argument N had an illegal value. (N < 0) = -3: The argument A() is invalid. = -4: The argument B() is invalid. = -5: The argument X() is invalid. = -8: The argument Nrhs had an illegal value. (Nrhs < 0) = i > 0: The i-th diagonal element of the factor U computed in double precision is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed. [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, where A is Hermitian positive definite and ( 2.20 -0.11+0.93i 0.81-0.37i ) A = ( -0.11-0.93i 2.32 -0.80+0.92i ) ( 0.81+0.37i -0.80-0.92i 2.29 ) ( 1.5980+1.4644i ) B = ( 1.3498+1.4398i ) ( 2.0561-0.5441i ) Sub Ex_Zcposv() Const N As Long = 3 Dim A(N - 1, N - 1) As Complex, B(N - 1) As Complex, X(N - 1) As Complex Dim Iter As Long, Info As Long A(0, 0) = Cmplx(2.2, 0) A(1, 0) = Cmplx(-0.11, -0.93): A(1, 1) = Cmplx(2.32, 0) A(2, 0) = Cmplx(0.81, 0.37): A(2, 1) = Cmplx(-0.8, -0.92): A(2, 2) = Cmplx(2.29, 0) B(0) = Cmplx(1.598, 1.4644): B(1) = Cmplx(1.3498, 1.4398): B(2) = Cmplx(2.0561, -0.5441) Call Zcposv("L", N, A(), B(), X(), Iter, 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 "Iter =", Iter, "Info =", Info End Sub Example Results X = 0.86 0.64 0.51 0.71 0.59 -0.15 Iter = 2 Info = 0