Zpbsv Zptsv Zptsvx

Zpbsv() Sub Zpbsv ( Uplo As  String, N As  Long, Kd As  Long, Ab() As  Complex, B() As  Complex, Info As  Long, Optional Nrhs As  Long = 1  ) (Simple driver) Solution to system of linear equations AX = B for a Hermitian positive definite band matrix 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 band matrix, and X and B are N x Nrhs matrices. The Cholesky decomposition is used to factor A as A = U^H * U, if Uplo = "U", or A = L * L^H, if Uplo = "L", where U is an upper triangular band matrix, and L is a lower triangular band matrix, with the same number of super-diagonals or sub-diagonals as A. The factored form of A is then used to solve the system of equations A * X = B. 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] Kd Number of super-diagonals of the matrix A if Uplo = "U", or number of sub-diagonals if Uplo = "L". (Kd >= 0) [in,out] Ab() Array Ab(LAb1 - 1, LAb2 - 1) (LAb1 >= Kd + 1, LAb2 >= N) [in] N x N Hermitian positive definite band matrix A in Kd+1 x N symmetric band matrix form. Upper or lower part is to be stored in accordance with uplo. See below for further details. [out] If Info = 0, the triangular factor U or L from the Cholesky factorization A = U^H*U or A = L*L^H of the band matrix A, in the same storage format as A. [in,out] 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) [in] N x Nrhs right hand side matrix B. [out] If Info = 0, the N x Nrhs solution matrix X. [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 Kd had an illegal value. (Kd < 0) = -4: The argument Ab() is invalid. = -5: The argument B() is invalid. = -7: The argument Nrhs had an illegal value. (Nrhs < 0) = i > 0: The leading minor of order i of A is not positive definite, so the factorization could not be completed, and the solution has not been 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) Further Details The symmetric band matrix form is illustrated by the following example, when n = 6, kd = 2, and uplo = "U": On entry: * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 On exit: * * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66 Similarly, if uplo = "L" the format of A is as follows: On entry: a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * * On exit: l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * * Array elements marked * are not used by the routine. Reference LAPACK Example Program Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where ( 2.88 0.29-0.44i 0 ) A = ( 0.29+0.44i 0.62 -0.01-0.02i ) ( 0 -0.01+0.02i 0.46 ) ( 1.6236-0.7300i ) B = ( 0.1581+0.1537i ) ( 0.1132-0.2290i ) Sub Ex_Zpbsv() Const N As Long = 3, Kd = 1 Dim Ab(Kd, N - 1) As Complex, B(N - 1) As Complex Dim ANorm As Double, RCond As Double, Info As Long Ab(0, 0) = cmplx(2.88): Ab(0, 1) = cmplx(0.62): Ab(0, 2) = cmplx(0.46) Ab(1, 0) = cmplx(0.29, 0.44): Ab(1, 1) = cmplx(-0.01, 0.02) B(0) = cmplx(1.6236, -0.73): B(1) = cmplx(0.1581, 0.1537): B(2) = cmplx(0.1132, -0.229) ANorm = Zlanhb("1", "L", N, Kd, Ab()) Call Zpbsv("L", N, Kd, Ab(), B(), Info) If Info = 0 Then Call Zpbcon("L", N, Kd, Ab(), 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 = 0.59 -0.28 -0.2 -0.04 0.24 -0.49 RCond = 0.124521368143895 Info = 0