SUBROUTINE ZGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
$ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
$ RCOND, FERR, BERR, WORK, RWORK, INFO )
*
* -- LAPACK driver routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER EQUED, FACT, TRANS
INTEGER INFO, KL, KU, LDAB, LDAFB, LDB, LDX, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), C( * ), FERR( * ), R( * ),
$ RWORK( * )
COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
* ..
*
* Purpose
* =======
*
* ZGBSVX uses the LU factorization to compute the solution to a complex
* 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 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.
*
* 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 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 (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 (Conjugate 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) COMPLEX*16 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) COMPLEX*16 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 ZGBTRF. 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 (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 factorization A = L*U
* as computed by ZGBTRF; 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 (input or 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 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) COMPLEX*16 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) COMPLEX*16 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 (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.
*
* 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).
*
* WORK (workspace) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace/output) DOUBLE PRECISION array, dimension (N)
* On exit, RWORK(1) contains the reciprocal pivot growth
* factor norm(A)/norm(U). The "max absolute element" norm is
* used. If RWORK(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 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.
*
* =====================================================================
* Moved setting of INFO = N+1 so INFO does not subsequently get
* overwritten. Sven, 17 Mar 05.
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL COLEQU, EQUIL, NOFACT, NOTRAN, ROWEQU
CHARACTER NORM
INTEGER I, INFEQU, J, J1, J2
DOUBLE PRECISION AMAX, ANORM, BIGNUM, COLCND, RCMAX, RCMIN,
$ ROWCND, RPVGRW, SMLNUM
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, ZLANGB, ZLANTB
EXTERNAL LSAME, DLAMCH, ZLANGB, ZLANTB
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZCOPY, ZGBCON, ZGBEQU, ZGBRFS, ZGBTRF,
$ ZGBTRS, ZLACPY, ZLAQGB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
NOTRAN = LSAME( TRANS, 'N' )
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
ROWEQU = .FALSE.
COLEQU = .FALSE.
ELSE
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
SMLNUM = DLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
END IF
*
* Test the input parameters.
*
IF( .NOT.NOFACT .AND. .NOT.EQUIL .AND. .NOT.LSAME( FACT, 'F' ) )
$ THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( KL.LT.0 ) THEN
INFO = -4
ELSE IF( KU.LT.0 ) THEN
INFO = -5
ELSE IF( NRHS.LT.0 ) THEN
INFO = -6
ELSE IF( LDAB.LT.KL+KU+1 ) THEN
INFO = -8
ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN
INFO = -10
ELSE IF( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( ROWEQU .OR. COLEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -12
ELSE
IF( ROWEQU ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 10 J = 1, N
RCMIN = MIN( RCMIN, R( J ) )
RCMAX = MAX( RCMAX, R( J ) )
10 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -13
ELSE IF( N.GT.0 ) THEN
ROWCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
ROWCND = ONE
END IF
END IF
IF( COLEQU .AND. INFO.EQ.0 ) THEN
RCMIN = BIGNUM
RCMAX = ZERO
DO 20 J = 1, N
RCMIN = MIN( RCMIN, C( J ) )
RCMAX = MAX( RCMAX, C( J ) )
20 CONTINUE
IF( RCMIN.LE.ZERO ) THEN
INFO = -14
ELSE IF( N.GT.0 ) THEN
COLCND = MAX( RCMIN, SMLNUM ) / MIN( RCMAX, BIGNUM )
ELSE
COLCND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -16
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -18
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGBSVX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL ZGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL ZLAQGB( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
$ AMAX, EQUED )
ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' )
COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' )
END IF
END IF
*
* Scale the right hand side.
*
IF( NOTRAN ) THEN
IF( ROWEQU ) THEN
DO 40 J = 1, NRHS
DO 30 I = 1, N
B( I, J ) = R( I )*B( I, J )
30 CONTINUE
40 CONTINUE
END IF
ELSE IF( COLEQU ) THEN
DO 60 J = 1, NRHS
DO 50 I = 1, N
B( I, J ) = C( I )*B( I, J )
50 CONTINUE
60 CONTINUE
END IF
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the LU factorization of the band matrix A.
*
DO 70 J = 1, N
J1 = MAX( J-KU, 1 )
J2 = MIN( J+KL, N )
CALL ZCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
$ AFB( KL+KU+1-J+J1, J ), 1 )
70 CONTINUE
*
CALL ZGBTRF( N, N, KL, KU, AFB, LDAFB, IPIV, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
ANORM = ZERO
DO 90 J = 1, INFO
DO 80 I = MAX( KU+2-J, 1 ), MIN( N+KU+1-J, KL+KU+1 )
ANORM = MAX( ANORM, ABS( AB( I, J ) ) )
80 CONTINUE
90 CONTINUE
RPVGRW = ZLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
$ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
$ RWORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = ANORM / RPVGRW
END IF
RWORK( 1 ) = RPVGRW
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A and the
* reciprocal pivot growth factor RPVGRW.
*
IF( NOTRAN ) THEN
NORM = '1'
ELSE
NORM = 'I'
END IF
ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK )
RPVGRW = ZLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, RWORK )
IF( RPVGRW.EQ.ZERO ) THEN
RPVGRW = ONE
ELSE
RPVGRW = ZLANGB( 'M', N, KL, KU, AB, LDAB, RWORK ) / RPVGRW
END IF
*
* Compute the reciprocal of the condition number of A.
*
CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
$ WORK, RWORK, INFO )
*
* Compute the solution matrix X.
*
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL ZGBTRS( TRANS, N, KL, KU, NRHS, AFB, LDAFB, IPIV, X, LDX,
$ INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL ZGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
$ B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO )
*
* Transform the solution matrix X to a solution of the original
* system.
*
IF( NOTRAN ) THEN
IF( COLEQU ) THEN
DO 110 J = 1, NRHS
DO 100 I = 1, N
X( I, J ) = C( I )*X( I, J )
100 CONTINUE
110 CONTINUE
DO 120 J = 1, NRHS
FERR( J ) = FERR( J ) / COLCND
120 CONTINUE
END IF
ELSE IF( ROWEQU ) THEN
DO 140 J = 1, NRHS
DO 130 I = 1, N
X( I, J ) = R( I )*X( I, J )
130 CONTINUE
140 CONTINUE
DO 150 J = 1, NRHS
FERR( J ) = FERR( J ) / ROWCND
150 CONTINUE
END IF
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
RWORK( 1 ) = RPVGRW
RETURN
*
* End of ZGBSVX
*
END