DGBSVX 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.

 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 triangular


    matrices with KL subdiagonals, and U is upper triangular with


    KL+KU superdiagonals.


 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.

FACT

          FACT is 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 modified.


          = 'N':  The matrix A will be copied to AFB and factored.


          = 'E':  The matrix A will be equilibrated if necessary, then


                  copied to AFB and factored.

TRANS

          TRANS is 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

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

KL

          KL is INTEGER


          The number of subdiagonals within the band of A.  KL >= 0.

KU

          KU is INTEGER


          The number of superdiagonals within the band of A.  KU >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

AB

          AB is DOUBLE PRECISION 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

          LDAB is INTEGER


          The leading dimension of the array AB.  LDAB >= KL+KU+1.

AFB

          AFB is DOUBLE PRECISION 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 DGBTRF.  U is stored as an upper triangular


          band matrix with KL+KU superdiagonals 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.  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

          LDAFB is INTEGER


          The leading dimension of the array AFB.  LDAFB >= 2*KL+KU+1.

IPIV

          IPIV is INTEGER array, dimension (N)


          If FACT = 'F', then IPIV is an input argument and on entry


          contains the pivot indices from the factorization A = L*U


          as computed by DGBTRF; 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 factorization 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 factorization A = L*U


          of the equilibrated matrix A.

EQUED

          EQUED is 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 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

          R is 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

          C is 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

          B is DOUBLE PRECISION 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

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, 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

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

RCOND

          RCOND is 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.

FERR

          FERR is 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

          BERR is 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).

WORK

          WORK is DOUBLE PRECISION array, dimension (3*N)


          On exit, WORK(1) contains the reciprocal pivot growth


          factor norm(A)/norm(U). The "max absolute element" norm is


          used. If WORK(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


          WORK(1) contains the reciprocal pivot growth factor for the


          leading INFO columns of A.

IWORK

          IWORK is INTEGER array, dimension (N)

INFO

          INFO is 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.

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April 2012