Purpose ======= LA_GBSVX computes the solution to a real or complex linear system of equations of the form A*X = B, A^T*X = B or A^H*X = B, where A is a square band matrix and X and B are rectangular matrices or vectors. LA_GBSVX can also optionally equilibrate the system if A is poorly scaled, estimate the condition number of (the equilibrated) A, return the pivot growth factor, and compute error bounds. ========= SUBROUTINE LA_GBSVX( AB, B, X, KL=kl, AFB=afb, IPIV=ipiv, & FACT=fact, TRANS=trans, EQUED=equed, R=r, C=c, & FERR=ferr, BERR=berr, RCOND=rcond, & RPVGRW=rpvgrw, INFO=info ) (), INTENT(INOUT) :: AB(:,:), (), INTENT(OUT) :: INTEGER, INTENT(IN), OPTIONAL :: KL (), INTENT(INOUT), OPTIONAL :: AFB(:,:) INTEGER, INTENT(INOUT), OPTIONAL :: IPIV(:) CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: TRANS, FACT CHARACTER(LEN=1), INTENT(INOUT), OPTIONAL :: EQUED REAL(), INTENT(INOUT), OPTIONAL :: C(:), R(:) REAL(), INTENT(OUT), OPTIONAL :: , RCOND, RPVGRW INTEGER, INTENT(OUT), OPTIONAL :: INFO where ::= REAL | COMPLEX ::= KIND(1.0) | KIND(1.0D0) ::= B(:,:) | B(:) ::= X(:,:) | X(:) ::= FERR(:), BERR(:) | FERR, BERR Arguments ========= AB (input/output) REAL or COMPLEX rectangular array, shape (:,:) with size(AB,1) = kl + ku + 1 and size(AB,2) = n, where kl and ku are, respectively, the numbers of subdiagonals and superdiagonals in the band of A, and n is the order of A. On entry, the matrix A or its equilibration in band storage. The (kl + ku + 1) diagonals of A are stored in rows 1 to (kl + ku + 1) of AB, so that the j-th column of A is stored in the j-th column of AB as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl) 1<=j<=n. The remaining elements in AB need not be set. If FACT = 'F' and EQUED /= 'N' then A has been equilibrated by the scaling factors in R and/or C during the previous call to LA GBSVX. On exit, if FACT = 'E', the equilibrated version of A is stored in AB; otherwise, AB is unchanged. B (input/output) REAL or COMPLEX array, shape (:,:) with size(B,1) = n or shape (:) with size(B) = n. On entry, the matrix B. On exit, the scaled version of B if the system has been equilibrated; otherwise, B is unchanged. X (output) REAL or COMPLEX array, shape (:,:) with size(X,1)=n and size(X,2) = size(B,2), or shape (:) with size(X) = n. The solution matrix X . KL Optional (input) INTEGER. The number of subdiagonals in the band of A (KL = kl). The number of superdiagonals in the band is given by ku = size(AB,1) - kl - 1. Default value: (size(AB,1) - 1) / 2. AFB Optional (input or output) REAL or COMPLEX rectangular array, shape (:,:) with size(AFB,1) = 2*kl+ku+1 and size(AFB,2)=n If FACT = 'F' then AFB is an input argument that contains the details of the factorization of (the equilibrated) A returned by a previous call to LA_GBSVX. If FACT /= 'F' then AFB is an output argument that contains the details of the factorization of (the equilibrated) A. U is an upper triangular band matrix with (kl + ku + 1) diagonals. These are stored in the first (kl + ku + 1) rows of AFB. The multipliers that arise during the factorization are stored in the remaining rows. IPIV Optional (input or output) INTEGER array, shape (:) with size(IPIV) = n. If FACT = 'F' then IPIV is an input argument that contains the pivot indices from the factorization of (the equilibrated) A, returned by a previous call to LA_GBSVX. If FACT /= 'F' then IPIV is an output argument that contains the pivot indices from the factorization of (the equilibrated) A. FACT Optional (input) CHARACTER(LEN=1). Specifies whether the factored form of the matrix A is supplied on entry, and, if not, whether the matrix A should be equilibrated before it is factored. = 'N': The matrix A will be copied to AFB and factored (no equilibration). = 'E': The matrix A will be equilibrated, then copied to AFB and factored. = 'F': AFB and IPIV contain the factored form of (the equilibrated) A. Default value: 'N'. TRANS Optional (input) CHARACTER(LEN=1). Specifies the form of the system of equations: = 'N': A*X = B (No transpose) = 'T': A^T*X = B (Transpose) = 'C': A^H*X = B (Conjugate transpose) Default value: 'N'. EQUED Optional (input or output) CHARACTER(LEN=1). Specifies the form of equilibration that was done. EQUED is an input argument if FACT = 'F', otherwise it is an output argument: = 'N': No equilibration (always true if FACT = 'N'). = 'R': Row equilibration, i.e., A has been premultiplied by diag(R). = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both row and column equilibration. Default value: 'N'. R Optional (input or output) REAL array, shape (:) with size(R) = size(A,1). The row scale factors for A. R is an input argument if FACT = 'F' and EQUED = 'R' or 'B'. R is an output argument if FACT = 'E' and EQUED = 'R' or 'B'. C Optional (input or output) REAL array, shape (:) with size(C) = size(A,1). The column scale factors for A. C is an input argument if FACT = 'F' and EQUED = 'C' or 'B'. C is an output argument if FACT = 'E' and EQUED = 'C' or 'B'. FERR Optional (output) REAL array of shape (:), with size(FERR) = size(X,2), or REAL scalar. The estimated 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) 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). The estimate is as reliable as the estimate for RCOND and is almost always a slight overestimate of the true error. BERR Optional (output) REAL array of shape (:), with size(BERR) = size(X,2), or REAL scalar. 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). RCOND Optional (output) REAL. The estimate of the reciprocal condition number of (the equilibrated) A. If RCOND is less than the machine precision, the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. RPVGRW Optional (output) REAL. The reciprocal pivot growth factor ||A||inf = ||U||inf. If RPVGRW is much less than 1, then the stability of the LU factorization of the (equilibrated) matrix A could be poor. This also means that the solution X , condition estimator RCOND, and forward error bound FERR could be unreliable. If the factorization fails with 0 < INFO <= size(A,1), then RPVGRW contains the reciprocal pivot growth factor for the leading INFO columns of A. INFO Optional (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, and i is <= n: U(i,i) = 0. The factorization has been completed, but the factor U is singular, so the solution could not be computed. = n+1: U is nonsingular, but RCOND is less than machine precision, so the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because the computed solution can be more accurate than the value of RCOND would suggest. If INFO is not present and an error occurs, then the program is terminated with an error message.