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NAME SGBSVX - use the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, SYNOPSIS SUBROUTINE SGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) CHARACTER EQUED, FACT, TRANS INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS REAL RCOND INTEGER IPIV( * ), IWORK( * ) REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), BERR( * ), C( * ), FERR( * ), R( * ), WORK( * ), X( LDX, * ) PURPOSE SGBSVX uses the LU factorization to compute the solution to a real system of linear equations A * X = B, A**T * X = B, or A**H * X = B, where A is a band matrix of order N with KL subdiagonals and KU superdiagonals, and X and B are N-by- NRHS matrices. Error bounds on the solution and a condition estimate are also provided. DESCRIPTION The following steps are performed by this subroutine: 1. If FACT = 'E', real scaling factors are computed to equilibrate the system: TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') or diag(C)*B (if TRANS = 'T' or 'C'). 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if FACT = 'E') as A = L * U, where L is a product of permutation and unit lower tri- angular matrices with KL subdiagonals, and U is upper triangular with KL+KU superdiagonals. 3. The factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, steps 4-6 are skipped. 4. The system of equations is solved for X using the fac- tored form of A. 5. Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it. 6. If equilibration was used, the matrix X is premultiplied by diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so that it solves the original system before equilibration. ARGUMENTS FACT (input) CHARACTER*1 Specifies whether or not 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. = 'F': On entry, AFB and IPIV contain the factored form of A. If EQUED is not 'N', the matrix A has been equilibrated with scaling factors given by R and C. AB, AFB, and IPIV are not modi- fied. = 'N': The matrix A will be copied to AFB and factored. = 'E': The matrix A will be equilibrated if neces- sary, then copied to AFB and factored. TRANS (input) CHARACTER*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 (Transpose) N (input) INTEGER The number of linear equations, i.e., the order of the matrix A. N >= 0. KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 0. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. AB (input/output) REAL array, dimension (LDAB,N) On entry, the matrix A in band storage, in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl) If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. AB is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .ne. 'N', A is scaled as follows: EQUED = 'R': A := diag(R) * A EQUED = 'C': A := A * diag(C) EQUED = 'B': A := diag(R) * A * diag(C). LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input or output) REAL array, dimension (LDAFB,N) If FACT = 'F', then AFB is an input argument and on entry contains details of the LU factorization of the band matrix A, as computed by SGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multi- pliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is the factored form of the equilibrated matrix A. If FACT = 'N', then AFB is an output argument and on exit returns details of the LU factorization of A. If FACT = 'E', then AFB is an output argument and on exit returns details of the LU factorization of the equilibrated matrix A (see the description of AB for the form of the equilibrated matrix). LDAFB (input) INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1. IPIV (input or output) INTEGER array, dimension (N) If FACT = 'F', then IPIV is an input argument and on entry contains the pivot indices from the factoriza- tion A = L*U as computed by SGBTRF; row i of the matrix was interchanged with row IPIV(i). If FACT = 'N', then IPIV is an output argument and on exit contains the pivot indices from the factori- zation A = L*U of the original matrix A. If FACT = 'E', then IPIV is an output argument and on exit contains the pivot indices from the factori- zation A = L*U of the equilibrated matrix A. EQUED (input/output) CHARACTER*1 Specifies the form of equilibration that was done. = 'N': No equilibration (always true if FACT = 'N'). = 'R': Row equilibration, i.e., A has been premul- tiplied by diag(R). = 'C': Column equilibration, i.e., A has been postmultiplied by diag(C). = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C). EQUED is an input variable if FACT = 'F'; otherwise, it is an output variable. R (input/output) REAL array, dimension (N) The row scale factors for A. If EQUED = 'R' or 'B', A is multiplied on the left by diag(R); if EQUED = 'N' or 'C', R is not accessed. R is an input vari- able if FACT = 'F'; otherwise, R is an output vari- able. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. C (input/output) REAL array, dimension (N) The column scale factors for A. If EQUED = 'C' or 'B', A is multiplied on the right by diag(C); if EQUED = 'N' or 'R', C is not accessed. C is an input variable if FACT = 'F'; otherwise, C is an output variable. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. B (input/output) REAL array, dimension (LDB,NRHS) On entry, the right hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is overwritten by diag(C)*B. LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) REAL array, dimension (LDX,NRHS) If INFO = 0, the N-by-NRHS solution matrix X to the original system of equations. Note that A and B are modified on exit if EQUED .ne. 'N', and the solution to the equilibrated system is inv(diag(C))*X if TRANS = 'N' and EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' and EQUED = 'R' or 'B'. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). RCOND (output) REAL The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in partic- ular, if RCOND = 0), the matrix is singular to work- ing precision. This condition is indicated by a return code of INFO > 0, and the solution and error bounds are not computed. FERR (output) REAL array, dimension (NRHS) The estimated forward error bounds for each solution vector X(j) (the j-th column of the solution matrix X). If XTRUE is the true solution, FERR(j) bounds the magnitude of the largest entry in (X(j) - XTRUE) divided by the magnitude of the largest entry in X(j). The quality of the error bound depends on the quality of the estimate of norm(inv(A)) computed in the code; if the estimate of norm(inv(A)) is accu- rate, the error bound is guaranteed. BERR (output) REAL array, dimension (NRHS) The componentwise relative backward error of each solution vector X(j) (i.e., the smallest relative change in any entry of A or B that makes X(j) an exact solution). WORK (workspace) REAL array, dimension (3*N) IWORK (workspace) INTEGER array, dimension (N) INFO (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) is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution and error bounds could not be computed. = N+1: RCOND is less than machine pre- cision. The factorization has been completed, but the matrix A is singular to working precision, and the solution and error bounds have not been com- puted.