**Purpose**

To solve (if well-conditioned) the matrix equations op( A )*X = B, where X and B are N-by-NRHS matrices, A is an N-by-N matrix and op( A ) is one of op( A ) = A or op( A ) = A'. Error bounds on the solution and a condition estimate are also provided.

SUBROUTINE MB02PD( FACT, TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, $ EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, $ IWORK, DWORK, INFO ) C .. Scalar Arguments .. CHARACTER EQUED, FACT, TRANS INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS DOUBLE PRECISION RCOND C .. Array Arguments .. INTEGER IPIV( * ), IWORK( * ) DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ), $ BERR( * ), C( * ), DWORK( * ), FERR( * ), $ R( * ), X( LDX, * )

**Mode Parameters**

FACT 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, AF 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. A, AF, and IPIV are not modified. = 'N': The matrix A will be copied to AF and factored. = 'E': The matrix A will be equilibrated if necessary, then copied to AF and factored. TRANS CHARACTER*1 Specifies the form of the system of equations as follows: = '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. NRHS (input) INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the leading N-by-N part of this array must contain the matrix A. If FACT = 'F' and EQUED is not 'N', then A must have been equilibrated by the scaling factors in R and/or C. A is not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit. On exit, if EQUED .NE. 'N', the leading N-by-N part of this array contains the matrix A scaled as follows: EQUED = 'R': A := diag(R) * A; EQUED = 'C': A := A * diag(C); EQUED = 'B': A := diag(R) * A * diag(C). LDA INTEGER The leading dimension of the array A. LDA >= max(1,N). AF (input or output) DOUBLE PRECISION array, dimension (LDAF,N) If FACT = 'F', then AF is an input argument and on entry the leading N-by-N part of this array must contain the factors L and U from the factorization A = P*L*U as computed by DGETRF. If EQUED .NE. 'N', then AF is the factored form of the equilibrated matrix A. If FACT = 'N', then AF is an output argument and on exit the leading N-by-N part of this array contains the factors L and U from the factorization A = P*L*U of the original matrix A. If FACT = 'E', then AF is an output argument and on exit the leading N-by-N part of this array contains the factors L and U from the factorization A = P*L*U of the equilibrated matrix A (see the description of A for the form of the equilibrated matrix). LDAF (input) INTEGER The leading dimension of the array AF. LDAF >= max(1,N). IPIV (input or output) INTEGER array, dimension (N) If FACT = 'F', then IPIV is an input argument and on entry it must contain the pivot indices from the factorization A = P*L*U as computed by DGETRF; row i of the matrix was interchanged with row IPIV(i). If FACT = 'N', then IPIV is an output argument and on exit it contains the pivot indices from the factorization A = P*L*U of the original matrix A. If FACT = 'E', then IPIV is an output argument and on exit it contains the pivot indices from the factorization A = P*L*U of the equilibrated matrix A. EQUED (input or output) CHARACTER*1 Specifies the form of equilibration that was done as follows: = '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, i.e., A has been replaced by diag(R) * A * diag(C). EQUED is an input argument if FACT = 'F'; otherwise, it is an output argument. R (input or output) DOUBLE PRECISION 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 argument if FACT = 'F'; otherwise, R is an output argument. If FACT = 'F' and EQUED = 'R' or 'B', each element of R must be positive. C (input or output) DOUBLE PRECISION 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 argument if FACT = 'F'; otherwise, C is an output argument. If FACT = 'F' and EQUED = 'C' or 'B', each element of C must be positive. B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the leading N-by-NRHS part of this array must contain the right-hand side matrix B. On exit, if EQUED = 'N', B is not modified; if TRANS = 'N' and EQUED = 'R' or 'B', the leading N-by-NRHS part of this array contains diag(R)*B; if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', the leading N-by-NRHS part of this array contains diag(C)*B. LDB INTEGER The leading dimension of the array B. LDB >= max(1,N). X (output) DOUBLE PRECISION array, dimension (LDX,NRHS) If INFO = 0 or INFO = N+1, the leading N-by-NRHS part of this array contains the 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) DOUBLE PRECISION The estimate of the reciprocal condition number of the matrix A after equilibration (if done). If RCOND is less than the machine precision (in particular, if RCOND = 0), the matrix is singular to working precision. This condition is indicated by a return code of INFO > 0. For efficiency reasons, RCOND is computed only when the matrix A is factored, i.e., for FACT = 'N' or 'E'. For FACT = 'F', RCOND is not used, but it is assumed that it has been computed and checked before the routine call. FERR (output) DOUBLE PRECISION array, dimension (NRHS) 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 (output) DOUBLE PRECISION array, dimension (NRHS) 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).

IWORK INTEGER array, dimension (N) DWORK DOUBLE PRECISION array, dimension (4*N) On exit, DWORK(1) contains the reciprocal pivot growth factor norm(A)/norm(U). The "max absolute element" norm is used. If DWORK(1) 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 factorization fails with 0 < INFO <= N, then DWORK(1) contains the reciprocal pivot growth factor for the leading INFO columns of A.

INFO 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. RCOND = 0 is returned. = N+1: U is nonsingular, but RCOND is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of RCOND would suggest. The positive values for INFO are set only when the matrix A is factored, i.e., for FACT = 'N' or 'E'.

The following steps are performed: 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 = P * L * U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular. 3. If some U(i,i)=0, so that U is exactly singular, then the routine returns with INFO = i. Otherwise, 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, INFO = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below. 4. The system of equations is solved for X using the factored 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.

[1] Anderson, E., Bai, Z., Bischof, C., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Ostrouchov, S., Sorensen, D. LAPACK Users' Guide: Second Edition, SIAM, Philadelphia, 1995.

This is a simplified version of the LAPACK Library routine DGESVX, useful when several sets of matrix equations with the same coefficient matrix A and/or A' should be solved.

3 The algorithm requires 0(N ) operations.

**Program Text**

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