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SUBROUTINE <a name="DGBSVX.1"></a><a href="dgbsvx.f.html#DGBSVX.1">DGBSVX</a>( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, IWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER IPIV( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ BERR( * ), C( * ), FERR( * ), R( * ),
$ WORK( * ), X( LDX, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DGBSVX.24"></a><a href="dgbsvx.f.html#DGBSVX.1">DGBSVX</a> uses the LU factorization to compute the solution to a real
</span><span class="comment">*</span><span class="comment"> system of linear equations A * X = B, A**T * X = B, or A**H * X = B,
</span><span class="comment">*</span><span class="comment"> where A is a band matrix of order N with KL subdiagonals and KU
</span><span class="comment">*</span><span class="comment"> superdiagonals, and X and B are N-by-NRHS matrices.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Error bounds on the solution and a condition estimate are also
</span><span class="comment">*</span><span class="comment"> provided.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Description
</span><span class="comment">*</span><span class="comment"> ===========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The following steps are performed by this subroutine:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 1. If FACT = 'E', real scaling factors are computed to equilibrate
</span><span class="comment">*</span><span class="comment"> the system:
</span><span class="comment">*</span><span class="comment"> TRANS = 'N': diag(R)*A*diag(C) *inv(diag(C))*X = diag(R)*B
</span><span class="comment">*</span><span class="comment"> TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B
</span><span class="comment">*</span><span class="comment"> TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B
</span><span class="comment">*</span><span class="comment"> Whether or not the system will be equilibrated depends on the
</span><span class="comment">*</span><span class="comment"> scaling of the matrix A, but if equilibration is used, A is
</span><span class="comment">*</span><span class="comment"> overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N')
</span><span class="comment">*</span><span class="comment"> or diag(C)*B (if TRANS = 'T' or 'C').
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor the
</span><span class="comment">*</span><span class="comment"> matrix A (after equilibration if FACT = 'E') as
</span><span class="comment">*</span><span class="comment"> A = L * U,
</span><span class="comment">*</span><span class="comment"> where L is a product of permutation and unit lower triangular
</span><span class="comment">*</span><span class="comment"> matrices with KL subdiagonals, and U is upper triangular with
</span><span class="comment">*</span><span class="comment"> KL+KU superdiagonals.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 3. If some U(i,i)=0, so that U is exactly singular, then the routine
</span><span class="comment">*</span><span class="comment"> returns with INFO = i. Otherwise, the factored form of A is used
</span><span class="comment">*</span><span class="comment"> to estimate the condition number of the matrix A. If the
</span><span class="comment">*</span><span class="comment"> reciprocal of the condition number is less than machine precision,
</span><span class="comment">*</span><span class="comment"> INFO = N+1 is returned as a warning, but the routine still goes on
</span><span class="comment">*</span><span class="comment"> to solve for X and compute error bounds as described below.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 4. The system of equations is solved for X using the factored form
</span><span class="comment">*</span><span class="comment"> of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 5. Iterative refinement is applied to improve the computed solution
</span><span class="comment">*</span><span class="comment"> matrix and calculate error bounds and backward error estimates
</span><span class="comment">*</span><span class="comment"> for it.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> 6. If equilibration was used, the matrix X is premultiplied by
</span><span class="comment">*</span><span class="comment"> diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so
</span><span class="comment">*</span><span class="comment"> that it solves the original system before equilibration.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> FACT (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Specifies whether or not the factored form of the matrix A is
</span><span class="comment">*</span><span class="comment"> supplied on entry, and if not, whether the matrix A should be
</span><span class="comment">*</span><span class="comment"> equilibrated before it is factored.
</span><span class="comment">*</span><span class="comment"> = 'F': On entry, AFB and IPIV contain the factored form of
</span><span class="comment">*</span><span class="comment"> A. If EQUED is not 'N', the matrix A has been
</span><span class="comment">*</span><span class="comment"> equilibrated with scaling factors given by R and C.
</span><span class="comment">*</span><span class="comment"> AB, AFB, and IPIV are not modified.
</span><span class="comment">*</span><span class="comment"> = 'N': The matrix A will be copied to AFB and factored.
</span><span class="comment">*</span><span class="comment"> = 'E': The matrix A will be equilibrated if necessary, then
</span><span class="comment">*</span><span class="comment"> copied to AFB and factored.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TRANS (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Specifies the form of the system of equations.
</span><span class="comment">*</span><span class="comment"> = 'N': A * X = B (No transpose)
</span><span class="comment">*</span><span class="comment"> = 'T': A**T * X = B (Transpose)
</span><span class="comment">*</span><span class="comment"> = 'C': A**H * X = B (Transpose)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of linear equations, i.e., the order of the
</span><span class="comment">*</span><span class="comment"> matrix A. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KL (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of subdiagonals within the band of A. KL >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KU (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of superdiagonals within the band of A. KU >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NRHS (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of right hand sides, i.e., the number of columns
</span><span class="comment">*</span><span class="comment"> of the matrices B and X. NRHS >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
</span><span class="comment">*</span><span class="comment"> On entry, the matrix A in band storage, in rows 1 to KL+KU+1.
</span><span class="comment">*</span><span class="comment"> The j-th column of A is stored in the j-th column of the
</span><span class="comment">*</span><span class="comment"> array AB as follows:
</span><span class="comment">*</span><span class="comment"> AB(KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+kl)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'F' and EQUED is not 'N', then A must have been
</span><span class="comment">*</span><span class="comment"> equilibrated by the scaling factors in R and/or C. AB is not
</span><span class="comment">*</span><span class="comment"> modified if FACT = 'F' or 'N', or if FACT = 'E' and
</span><span class="comment">*</span><span class="comment"> EQUED = 'N' on exit.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> On exit, if EQUED .ne. 'N', A is scaled as follows:
</span><span class="comment">*</span><span class="comment"> EQUED = 'R': A := diag(R) * A
</span><span class="comment">*</span><span class="comment"> EQUED = 'C': A := A * diag(C)
</span><span class="comment">*</span><span class="comment"> EQUED = 'B': A := diag(R) * A * diag(C).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDAB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array AB. LDAB >= KL+KU+1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AFB (input or output) DOUBLE PRECISION array, dimension (LDAFB,N)
</span><span class="comment">*</span><span class="comment"> If FACT = 'F', then AFB is an input argument and on entry
</span><span class="comment">*</span><span class="comment"> contains details of the LU factorization of the band matrix
</span><span class="comment">*</span><span class="comment"> A, as computed by <a name="DGBTRF.129"></a><a href="dgbtrf.f.html#DGBTRF.1">DGBTRF</a>. U is stored as an upper triangular
</span><span class="comment">*</span><span class="comment"> band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1,
</span><span class="comment">*</span><span class="comment"> and the multipliers used during the factorization are stored
</span><span class="comment">*</span><span class="comment"> in rows KL+KU+2 to 2*KL+KU+1. If EQUED .ne. 'N', then AFB is
</span><span class="comment">*</span><span class="comment"> the factored form of the equilibrated matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'N', then AFB is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> returns details of the LU factorization of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'E', then AFB is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> returns details of the LU factorization of the equilibrated
</span><span class="comment">*</span><span class="comment"> matrix A (see the description of AB for the form of the
</span><span class="comment">*</span><span class="comment"> equilibrated matrix).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDAFB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array AFB. LDAFB >= 2*KL+KU+1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> IPIV (input or output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> If FACT = 'F', then IPIV is an input argument and on entry
</span><span class="comment">*</span><span class="comment"> contains the pivot indices from the factorization A = L*U
</span><span class="comment">*</span><span class="comment"> as computed by <a name="DGBTRF.149"></a><a href="dgbtrf.f.html#DGBTRF.1">DGBTRF</a>; row i of the matrix was interchanged
</span><span class="comment">*</span><span class="comment"> with row IPIV(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'N', then IPIV is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> contains the pivot indices from the factorization A = L*U
</span><span class="comment">*</span><span class="comment"> of the original matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If FACT = 'E', then IPIV is an output argument and on exit
</span><span class="comment">*</span><span class="comment"> contains the pivot indices from the factorization A = L*U
</span><span class="comment">*</span><span class="comment"> of the equilibrated matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> EQUED (input or output) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> Specifies the form of equilibration that was done.
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