Dgbcon Dgbtrf Dgbtrs Dgecon Dgetrf Dgetri Dgetrs Dgtcon Dgttrf Dgttrs

Dgbtrf() Sub Dgbtrf ( M As  Long, N As  Long, Kl As  Long, Ku As  Long, Ab() As  Double, IPiv() As  Long, Info As  Long  ) LU factorization of a general band matrix Purpose This routine computes an LU factorization of a real m x n band matrix A using partial pivoting with row interchanges. This is the blocked version of the algorithm calling Level 3 BLAS. Parameters [in] M Number of rows of the matrix A. (M >= 0) (If M = 0, returns without computation) [in] N Number of columns of the matrix A. (N >= 0) (If N = 0, returns without computation) [in] Kl Number of subdiagonals within the band of A (Kl >= 0) [in] Ku Number of superdiagonals 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; for 1 <= i <= min(M, N), row i of the matrix was interchanged with row IPiv(i-1). [out] Info = 0: Successful exit. = -1: The argument M had an illegal value. (M < 0) = -2: The argument N had an illegal value. (N < 0) = -3: The argument Kl had an illegal value. (Kl < 0) = -4: The argument Ku had an illegal value. (Ku < 0) = -5: The argument Ab() is invalid. = -6: The argument IPiv() is invalid. = 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 division by zero will occur if it is used to solve a system of equations. Further Details The band matrix form is illustrated by the following example, when M = 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_Dgbtrf() 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, 0) = 0: 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: Ab(3, 2) = 0 B(0) = 0.7416: B(1) = 0.7885: B(2) = 1.0833 ANorm = Dlangb("1", N, Kl, Ku, Ab(), , Kl) Call Dgbtrf(N, N, Kl, Ku, Ab(), IPiv(), Info) If Info = 0 Then Call Dgbtrs("N", 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