Dgbsv Dgbsvx Dgesv Dgesvx Dgtsv Dgtsvx Dsgesv

Dgbsv() Sub Dgbsv ( N As  Long, Kl As  Long, Ku As  Long, Ab() As  Double, IPiv() As  Long, B() As  Double, Info As  Long, Optional Nrhs As  Long = 1  ) (Simple driver) Solution to system of linear equations AX = B for a general band matrix Purpose This routine computes the solution to a real system of linear equations A * X = B where A is a band matrix of order n with kl sub-diagonals and ku super-diagonals, and X and B are n x nrhs matrices. The LU decomposition with partial pivoting and row interchanges is used to factor A as A = L * U where L is a product of permutation and unit lower triangular matrices with kl sub-diagonals, and U is upper triangular with kl+ku super-diagonals. The factored form of A is then used to solve the system of equations A * X = B. Parameters [in] N Number of linear equations, i.e., order of the matrix A. (N >= 0) (If N = 0, returns without computation) [in] Kl Number of sub-diagonals within the band of A. (Kl >= 0) [in] Ku Number of super-diagonals within the band of A. (Ku >= 0) [in,out] Ab() Array Ab(LAb1 - 1, LAb2 - 1) (LAb1 >= 2Kl + Ku + 1, LAb2 >= N) [in] The matrix A in band matrix form, in rows Kl+1 to 2Kl+Ku+1; rows 1 to Kl of the array need not be set. [out] Details of the factorization: U is stored as an upper triangular band matrix with kl+ku super-diagonals in rows 1 to Kl+Ku+1, and the multipliers used during the factorization are stored in rows Kl+Ku+2 to 2*Kl+Ku+1. See below for further details. [out] IPiv() Array IPiv(LIPiv - 1) (LIPiv >= N) Pivot indices that define the permutation matrix P; row i of the matrix was interchanged with row IPiv(i-1). [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 matrix of right hand side matrix B. [out] If Info = 0, the N x Nrhs solution matrix X. [out] Info = 0: Successful exit. = -1: The argument N had an illegal value. (N < 0) = -2: The argument Kl had an illegal value. (Kl < 0) = -3: The argument Ku had an illegal value. (Ku < 0) = -4: The argument Ab() is invalid. = -5: The argument IPiv() is invalid. = -6: The argument B() is invalid. = -8: The argument Nrhs had an illegal value. (Nrhs < 0) = i > 0: The i-th diagonal element of the factor U is exactly zero. The factorization has been completed, but the factor U is exactly singular, 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 band matrix form is illustrated by the following example, when N = 6, Kl = 2, Ku = 1: On entry: * * * + + + * * + + + + * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * * On exit: * * * u14 u25 u36 * * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66 m21 m32 m43 m54 m65 * m31 m42 m53 m64 * * Array elements marked * are not used by the routine; elements marked + need not be set on entry, but are required by the routine to store elements of U because of fill-in resulting from the row interchanges. Reference LAPACK Example Program Solve the system of linear equations Ax = B and estimate the reciprocal of the condition number (RCond) of A, where ( 2.34 0.57 0 ) ( 0.7416 ) A = ( 0.65 1.98 -1.39 ), B = ( 0.7885 ) ( 0 1.50 1.73 ) ( 1.0833 ) Sub Ex_Dgbsv() Const N = 3, Kl = 1, Ku = 1 Dim Ab(2 * Kl + Ku, N - 1) As Double, B(N - 1) As Double, IPiv(N - 1) As Long Dim ANorm As Double, RCond As Double, Info As Long Ab(1, 1) = 0.57: Ab(1, 2) = -1.39 Ab(2, 0) = 2.34: Ab(2, 1) = 1.98: Ab(2, 2) = 1.73 Ab(3, 0) = 0.65: Ab(3, 1) = 1.5 B(0) = 0.7416: B(1) = 0.7885: B(2) = 1.0833 ANorm = Dlangb("1", N, Kl, Ku, Ab(), , Kl) Call Dgbsv(N, Kl, Ku, Ab(), IPiv(), B(), Info) If Info = 0 Then Call Dgbcon("1", N, Kl, Ku, Ab(), 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 = 0.2 0.48 0.21 RCond = 0.364187455306431 Info = 0