/R-2.15.1/src/modules/lapack/dlapack1.f
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- SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
- *
- * -- LAPACK driver routine (version 3.1) --
- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
- * November 2006
- *
- * .. Scalar Arguments ..
- INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
- * ..
- * .. Array Arguments ..
- INTEGER IPIV( * )
- DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
- * ..
- *
- * Purpose
- * =======
- *
- * DGBSV computes the solution to a real system of linear equations
- * A * 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.
- *
- * The LU decomposition with partial pivoting and row interchanges is
- * used to factor A 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. The factored form of A
- * is then used to solve the system of equations A * X = B.
- *
- * Arguments
- * =========
- *
- * 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 matrix B. NRHS >= 0.
- *
- * AB (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
- * On entry, the matrix A in band storage, in rows KL+1 to
- * 2*KL+KU+1; rows 1 to KL of the array need not be set.
- * The j-th column of A is stored in the j-th column of the
- * array AB as follows:
- * AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
- * On exit, details of the factorization: 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.
- * See below for further details.
- *
- * LDAB (input) INTEGER
- * The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
- *
- * IPIV (output) INTEGER array, dimension (N)
- * The pivot indices that define the permutation matrix P;
- * row i of the matrix was interchanged with row IPIV(i).
- *
- * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
- * On entry, the N-by-NRHS right hand side matrix B.
- * On exit, if INFO = 0, the N-by-NRHS solution matrix X.
- *
- * LDB (input) INTEGER
- * The leading dimension of the array B. LDB >= max(1,N).
- *
- * INFO (output) INTEGER
- * = 0: successful exit
- * < 0: if INFO = -i, the i-th argument had an illegal value
- * > 0: if INFO = i, U(i,i) is exactly zero. The factorization
- * has been completed, but the factor U is exactly
- * singular, and the solution has not been computed.
- *
- * Further Details
- * ===============
- *
- * The band storage scheme is illustrated by the following example, when
- * M = N = 6, KL = 2, KU = 1:
- *
- * On entry: On exit:
- *
- * * * * + + + * * * u14 u25 u36
- * * * + + + + * * u13 u24 u35 u46
- * * a12 a23 a34 a45 a56 * u12 u23 u34 u45 u56
- * a11 a22 a33 a44 a55 a66 u11 u22 u33 u44 u55 u66
- * a21 a32 a43 a54 a65 * m21 m32 m43 m54 m65 *
- * a31 a42 a53 a64 * * m31 m42 m53 m64 * *
- *
- * Array elements marked * are not used by the routine; elements marked
- * + need not be set on entry, but are required by the routine to store
- * elements of U because of fill-in resulting from the row interchanges.
- *
- * =====================================================================
- *
- * .. External Subroutines ..
- EXTERNAL DGBTRF, DGBTRS, XERBLA
- * ..
- * .. Intrinsic Functions ..
- INTRINSIC MAX
- * ..
- * .. Executable Statements ..
- *
- * Test the input parameters.
- *
- INFO = 0
- IF( N.LT.0 ) THEN
- INFO = -1
- ELSE IF( KL.LT.0 ) THEN
- INFO = -2
- ELSE IF( KU.LT.0 ) THEN
- INFO = -3
- ELSE IF( NRHS.LT.0 ) THEN
- INFO = -4
- ELSE IF( LDAB.LT.2*KL+KU+1 ) THEN
- INFO = -6
- ELSE IF( LDB.LT.MAX( N, 1 ) ) THEN
- INFO = -9
- END IF
- IF( INFO.NE.0 ) THEN
- CALL XERBLA( 'DGBSV ', -INFO )
- RETURN
- END IF
- *
- * Compute the LU factorization of the band matrix A.
- *
- CALL DGBTRF( N, N, KL, KU, AB, LDAB, IPIV, INFO )
- IF( INFO.EQ.0 ) THEN
- *
- * Solve the system A*X = B, overwriting B with X.
- *
- CALL DGBTRS( 'No transpose', N, KL, KU, NRHS, AB, LDAB, IPIV,
- $ B, LDB, INFO )
- END IF
- RETURN
- *
- * End of DGBSV
- *
- END
- SUBROUTINE DGBSVX( FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB,
- $ LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX,
- $ RCOND, FERR, BERR, WORK, IWORK, 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( * ), IWORK( * )
- DOUBLE PRECISION AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
- $ BERR( * ), C( * ), FERR( * ), R( * ),
- $ WORK( * ), X( LDX, * )
- * ..
- *
- * Purpose
- * =======
- *
- * 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.
- *
- * 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 (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) 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 (input) INTEGER
- * The leading dimension of the array AB. LDAB >= KL+KU+1.
- *
- * AFB (input or output) 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 (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 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 (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) 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 (input) 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 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/output) 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 (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. 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.
- *
- * =====================================================================
- *
- * .. 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, DLANGB, DLANTB
- EXTERNAL LSAME, DLAMCH, DLANGB, DLANTB
- * ..
- * .. External Subroutines ..
- EXTERNAL DCOPY, DGBCON, DGBEQU, DGBRFS, DGBTRF, DGBTRS,
- $ DLACPY, DLAQGB, XERBLA
- * ..
- * .. 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( 'DGBSVX', -INFO )
- RETURN
- END IF
- *
- IF( EQUIL ) THEN
- *
- * Compute row and column scalings to equilibrate the matrix A.
- *
- CALL DGBEQU( N, N, KL, KU, AB, LDAB, R, C, ROWCND, COLCND,
- $ AMAX, INFEQU )
- IF( INFEQU.EQ.0 ) THEN
- *
- * Equilibrate the matrix.
- *
- CALL DLAQGB( 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 DCOPY( J2-J1+1, AB( KU+1-J+J1, J ), 1,
- $ AFB( KL+KU+1-J+J1, J ), 1 )
- 70 CONTINUE
- *
- CALL DGBTRF( 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 = DLANTB( 'M', 'U', 'N', INFO, MIN( INFO-1, KL+KU ),
- $ AFB( MAX( 1, KL+KU+2-INFO ), 1 ), LDAFB,
- $ WORK )
- IF( RPVGRW.EQ.ZERO ) THEN
- RPVGRW = ONE
- ELSE
- RPVGRW = ANORM / RPVGRW
- END IF
- WORK( 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 = DLANGB( NORM, N, KL, KU, AB, LDAB, WORK )
- RPVGRW = DLANTB( 'M', 'U', 'N', N, KL+KU, AFB, LDAFB, WORK )
- IF( RPVGRW.EQ.ZERO ) THEN
- RPVGRW = ONE
- ELSE
- RPVGRW = DLANGB( 'M', N, KL, KU, AB, LDAB, WORK ) / RPVGRW
- END IF
- *
- * Compute the reciprocal of the condition number of A.
- *
- CALL DGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND,
- $ WORK, IWORK, INFO )
- *
- * Compute the solution matrix X.
- *
- CALL DLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
- CALL DGBTRS( 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 DGBRFS( TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV,
- $ B, LDB, X, LDX, FERR, BERR, WORK, IWORK, 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
- *
- WORK( 1 ) = RPVGRW
- RETURN
- *
- * End of DGBSVX
- *
- END
- SUBROUTINE DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
- $ VS, LDVS, WORK, LWORK, BWORK, INFO )
- *
- * -- LAPACK driver routine (version 3.1) --
- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
- * November 2006
- *
- * .. Scalar Arguments ..
- CHARACTER JOBVS, SORT
- INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
- * ..
- * .. Array Arguments ..
- LOGICAL BWORK( * )
- DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
- $ WR( * )
- * ..
- * .. Function Arguments ..
- LOGICAL SELECT
- EXTERNAL SELECT
- * ..
- *
- * Purpose
- * =======
- *
- * DGEES computes for an N-by-N real nonsymmetric matrix A, the
- * eigenvalues, the real Schur form T, and, optionally, the matrix of
- * Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
- *
- * Optionally, it also orders the eigenvalues on the diagonal of the
- * real Schur form so that selected eigenvalues are at the top left.
- * The leading columns of Z then form an orthonormal basis for the
- * invariant subspace corresponding to the selected eigenvalues.
- *
- * A matrix is in real Schur form if it is upper quasi-triangular with
- * 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
- * form
- * [ a b ]
- * [ c a ]
- *
- * where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
- *
- * Arguments
- * =========
- *
- * JOBVS (input) CHARACTER*1
- * = 'N': Schur vectors are not computed;
- * = 'V': Schur vectors are computed.
- *
- * SORT (input) CHARACTER*1
- * Specifies whether or not to order the eigenvalues on the
- * diagonal of the Schur form.
- * = 'N': Eigenvalues are not ordered;
- * = 'S': Eigenvalues are ordered (see SELECT).
- *
- * SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
- * SELECT must be declared EXTERNAL in the calling subroutine.
- * If SORT = 'S', SELECT is used to select eigenvalues to sort
- * to the top left of the Schur form.
- * If SORT = 'N', SELECT is not referenced.
- * An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
- * SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
- * conjugate pair of eigenvalues is selected, then both complex
- * eigenvalues are selected.
- * Note that a selected complex eigenvalue may no longer
- * satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
- * ordering may change the value of complex eigenvalues
- * (especially if the eigenvalue is ill-conditioned); in this
- * case INFO is set to N+2 (see INFO below).
- *
- * N (input) INTEGER
- * The order of the matrix A. N >= 0.
- *
- * A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
- * On entry, the N-by-N matrix A.
- * On exit, A has been overwritten by its real Schur form T.
- *
- * LDA (input) INTEGER
- * The leading dimension of the array A. LDA >= max(1,N).
- *
- * SDIM (output) INTEGER
- * If SORT = 'N', SDIM = 0.
- * If SORT = 'S', SDIM = number of eigenvalues (after sorting)
- * for which SELECT is true. (Complex conjugate
- * pairs for which SELECT is true for either
- * eigenvalue count as 2.)
- *
- * WR (output) DOUBLE PRECISION array, dimension (N)
- * WI (output) DOUBLE PRECISION array, dimension (N)
- * WR and WI contain the real and imaginary parts,
- * respectively, of the computed eigenvalues in the same order
- * that they appear on the diagonal of the output Schur form T.
- * Complex conjugate pairs of eigenvalues will appear
- * consecutively with the eigenvalue having the positive
- * imaginary part first.
- *
- * VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
- * If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
- * vectors.
- * If JOBVS = 'N', VS is not referenced.
- *
- * LDVS (input) INTEGER
- * The leading dimension of the array VS. LDVS >= 1; if
- * JOBVS = 'V', LDVS >= N.
- *
- * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
- * On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
- *
- * LWORK (input) INTEGER
- * The dimension of the array WORK. LWORK >= max(1,3*N).
- * For good performance, LWORK must generally be larger.
- *
- * If LWORK = -1, then a workspace query is assumed; the routine
- * only calculates the optimal size of the WORK array, returns
- * this value as the first entry of the WORK array, and no error
- * message related to LWORK is issued by XERBLA.
- *
- * BWORK (workspace) LOGICAL array, dimension (N)
- * Not referenced if SORT = '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: the QR algorithm failed to compute all the
- * eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
- * contain those eigenvalues which have converged; if
- * JOBVS = 'V', VS contains the matrix which reduces A
- * to its partially converged Schur form.
- * = N+1: the eigenvalues could not be reordered because some
- * eigenvalues were too close to separate (the problem
- * is very ill-conditioned);
- * = N+2: after reordering, roundoff changed values of some
- * complex eigenvalues so that leading eigenvalues in
- * the Schur form no longer satisfy SELECT=.TRUE. This
- * could also be caused by underflow due to scaling.
- *
- * =====================================================================
- *
- * .. Parameters ..
- DOUBLE PRECISION ZERO, ONE
- PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
- * ..
- * .. Local Scalars ..
- LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTST,
- $ WANTVS
- INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
- $ IHI, ILO, INXT, IP, ITAU, IWRK, MAXWRK, MINWRK
- DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, S, SEP, SMLNUM
- * ..
- * .. Local Arrays ..
- INTEGER IDUM( 1 )
- DOUBLE PRECISION DUM( 1 )
- * ..
- * .. External Subroutines ..
- EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
- $ DLABAD, DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
- * ..
- * .. External Functions ..
- LOGICAL LSAME
- INTEGER ILAENV
- DOUBLE PRECISION DLAMCH, DLANGE
- EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
- * ..
- * .. Intrinsic Functions ..
- INTRINSIC MAX, SQRT
- * ..
- * .. Executable Statements ..
- *
- * Test the input arguments
- *
- INFO = 0
- LQUERY = ( LWORK.EQ.-1 )
- WANTVS = LSAME( JOBVS, 'V' )
- WANTST = LSAME( SORT, 'S' )
- IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
- INFO = -1
- ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
- INFO = -2
- ELSE IF( N.LT.0 ) THEN
- INFO = -4
- ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
- INFO = -6
- ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN
- INFO = -11
- END IF
- *
- * Compute workspace
- * (Note: Comments in the code beginning "Workspace:" describe the
- * minimal amount of workspace needed at that point in the code,
- * as well as the preferred amount for good performance.
- * NB refers to the optimal block size for the immediately
- * following subroutine, as returned by ILAENV.
- * HSWORK refers to the workspace preferred by DHSEQR, as
- * calculated below. HSWORK is computed assuming ILO=1 and IHI=N,
- * the worst case.)
- *
- IF( INFO.EQ.0 ) THEN
- IF( N.EQ.0 ) THEN
- MINWRK = 1
- MAXWRK = 1
- ELSE
- MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 )
- MINWRK = 3*N
- *
- CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS,
- $ WORK, -1, IEVAL )
- HSWORK = WORK( 1 )
- *
- IF( .NOT.WANTVS ) THEN
- MAXWRK = MAX( MAXWRK, N + HSWORK )
- ELSE
- MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1,
- $ 'DORGHR', ' ', N, 1, N, -1 ) )
- MAXWRK = MAX( MAXWRK, N + HSWORK )
- END IF
- END IF
- WORK( 1 ) = MAXWRK
- *
- IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
- INFO = -13
- END IF
- END IF
- *
- IF( INFO.NE.0 ) THEN
- CALL XERBLA( 'DGEES ', -INFO )
- RETURN
- ELSE IF( LQUERY ) THEN
- RETURN
- END IF
- *
- * Quick return if possible
- *
- IF( N.EQ.0 ) THEN
- SDIM = 0
- RETURN
- END IF
- *
- * Get machine constants
- *
- EPS = DLAMCH( 'P' )
- SMLNUM = DLAMCH( 'S' )
- BIGNUM = ONE / SMLNUM
- CALL DLABAD( SMLNUM, BIGNUM )
- SMLNUM = SQRT( SMLNUM ) / EPS
- BIGNUM = ONE / SMLNUM
- *
- * Scale A if max element outside range [SMLNUM,BIGNUM]
- *
- ANRM = DLANGE( 'M', N, N, A, LDA, DUM )
- SCALEA = .FALSE.
- IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
- SCALEA = .TRUE.
- CSCALE = SMLNUM
- ELSE IF( ANRM.GT.BIGNUM ) THEN
- SCALEA = .TRUE.
- CSCALE = BIGNUM
- END IF
- IF( SCALEA )
- $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR )
- *
- * Permute the matrix to make it more nearly triangular
- * (Workspace: need N)
- *
- IBAL = 1
- CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR )
- *
- * Reduce to upper Hessenberg form
- * (Workspace: need 3*N, prefer 2*N+N*NB)
- *
- ITAU = N + IBAL
- IWRK = N + ITAU
- CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ),
- $ LWORK-IWRK+1, IERR )
- *
- IF( WANTVS ) THEN
- *
- * Copy Householder vectors to VS
- *
- CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS )
- *
- * Generate orthogonal matrix in VS
- * (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB)
- *
- CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ),
- $ LWORK-IWRK+1, IERR )
- END IF
- *
- SDIM = 0
- *
- * Perform QR iteration, accumulating Schur vectors in VS if desired
- * (Workspace: need N+1, prefer N+HSWORK (see comments) )
- *
- IWRK = ITAU
- CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS,
- $ WORK( IWRK ), LWORK-IWRK+1, IEVAL )
- IF( IEVAL.GT.0 )
- $ INFO = IEVAL
- *
- * Sort eigenvalues if desired
- *
- IF( WANTST .AND. INFO.EQ.0 ) THEN
- IF( SCALEA ) THEN
- CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR )
- CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR )
- END IF
- DO 10 I = 1, N
- BWORK( I ) = SELECT( WR( I ), WI( I ) )
- 10 CONTINUE
- *
- * Reorder eigenvalues and transform Schur vectors
- * (Workspace: none needed)
- *
- CALL DTRSEN( 'N', JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI,
- $ SDIM, S, SEP, WORK( IWRK ), LWORK-IWRK+1, IDUM, 1,
- $ ICOND )
- IF( ICOND.GT.0 )
- $ INFO = N + ICOND
- END IF
- *
- IF( WANTVS ) THEN
- *
- * Undo balancing
- * (Workspace: need N)
- *
- CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS,
- $ IERR )
- END IF
- *
- IF( SCALEA ) THEN
- *
- * Undo scaling for the Schur form of A
- *
- CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR )
- CALL DCOPY( N, A, LDA+1, WR, 1 )
- IF( CSCALE.EQ.SMLNUM ) THEN
- *
- * If scaling back towards underflow, adjust WI if an
- * offdiagonal element of a 2-by-2 block in the Schur form
- * underflows.
- *
- IF( IEVAL.GT.0 ) THEN
- I1 = IEVAL + 1
- I2 = IHI - 1
- CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI,
- $ MAX( ILO-1, 1 ), IERR )
- ELSE IF( WANTST ) THEN
- I1 = 1
- I2 = N - 1
- ELSE
- I1 = ILO
- I2 = IHI - 1
- END IF
- INXT = I1 - 1
- DO 20 I = I1, I2
- IF( I.LT.INXT )
- $ GO TO 20
- IF( WI( I ).EQ.ZERO ) THEN
- INXT = I + 1
- ELSE
- IF( A( I+1, I ).EQ.ZERO ) THEN
- WI( I ) = ZERO
- WI( I+1 ) = ZERO
- ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ.
- $ ZERO ) THEN
- WI( I ) = ZERO
- WI( I+1 ) = ZERO
- IF( I.GT.1 )
- $ CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 )
- IF( N.GT.I+1 )
- $ CALL DSWAP( N-I-1, A( I, I+2 ), LDA,
- $ A( I+1, I+2 ), LDA )
- IF( WANTVS ) THEN
- CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 )
- END IF
- A( I, I+1 ) = A( I+1, I )
- A( I+1, I ) = ZERO
- END IF
- INXT = I + 2
- END IF
- 20 CONTINUE
- END IF
- *
- * Undo scaling for the imaginary part of the eigenvalues
- *
- CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1,
- $ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR )
- END IF
- *
- IF( WANTST .AND. INFO.EQ.0 ) THEN
- *
- * Check if reordering successful
- *
- LASTSL = .TRUE.
- LST2SL = .TRUE.
- SDIM = 0
- IP = 0
- DO 30 I = 1, N
- CURSL = SELECT( WR( I ), WI( I ) )
- IF( WI( I ).EQ.ZERO ) THEN
- IF( CURSL )
- $ SDIM = SDIM + 1
- IP = 0
- IF( CURSL .AND. .NOT.LASTSL )
- $ INFO = N + 2
- ELSE
- IF( IP.EQ.1 ) THEN
- *
- * Last eigenvalue of conjugate pair
- *
- CURSL = CURSL .OR. LASTSL
- LASTSL = CURSL
- IF( CURSL )
- $ SDIM = SDIM + 2
- IP = -1
- IF( CURSL .AND. .NOT.LST2SL )
- $ INFO = N + 2
- ELSE
- *
- * First eigenvalue of conjugate pair
- *
- IP = 1
- END IF
- END IF
- LST2SL = LASTSL
- LASTSL = CURSL
- 30 CONTINUE
- END IF
- *
- WORK( 1 ) = MAXWRK
- RETURN
- *
- * End of DGEES
- *
- END
- SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM,
- $ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK,
- $ IWORK, LIWORK, BWORK, INFO )
- *
- * -- LAPACK driver routine (version 3.1) --
- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
- * November 2006
- *
- * .. Scalar Arguments ..
- CHARACTER JOBVS, SENSE, SORT
- INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
- DOUBLE PRECISION RCONDE, RCONDV
- * ..
- * .. Array Arguments ..
- LOGICAL BWORK( * )
- INTEGER IWORK( * )
- DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
- $ WR( * )
- * ..
- * .. Function Arguments ..
- LOGICAL SELECT
- EXTERNAL SELECT
- * ..
- *
- * Purpose
- * =======
- *
- * DGEESX computes for an N-by-N real nonsymmetric matrix A, the
- * eigenvalues, the real Schur form T, and, optionally, the matrix of
- * Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T).
- *
- * Optionally, it also orders the eigenvalues on the diagonal of the
- * real Schur form so that selected eigenvalues are at the top left;
- * computes a reciprocal condition number for the average of the
- * selected eigenvalues (RCONDE); and computes a reciprocal condition
- * number for the right invariant subspace corresponding to the
- * selected eigenvalues (RCONDV). The leading columns of Z form an
- * orthonormal basis for this invariant subspace.
- *
- * For further explanation of the reciprocal condition numbers RCONDE
- * and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where
- * these quantities are called s and sep respectively).
- *
- * A real matrix is in real Schur form if it is upper quasi-triangular
- * with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
- * the form
- * [ a b ]
- * [ c a ]
- *
- * where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
- *
- * Arguments
- * =========
- *
- * JOBVS (input) CHARACTER*1
- * = 'N': Schur vectors are not computed;
- * = 'V': Schur vectors are computed.
- *
- * SORT (input) CHARACTER*1
- * Specifies whether or not to order the eigenvalues on the
- * diagonal of the Schur form.
- * = 'N': Eigenvalues are not ordered;
- * = 'S': Eigenvalues are ordered (see SELECT).
- *
- * SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
- * SELECT must be declared EXTERNAL in the calling subroutine.
- * If SORT = 'S', SELECT is used to select eigenvalues to sort
- * to the top left of the Schur form.
- * If SORT = 'N', SELECT is not referenced.
- * An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
- * SELECT(WR(j),WI(j)) is true; i.e., if either one of a
- * complex conjugate pair of eigenvalues is selected, then both
- * are. Note that a selected complex eigenvalue may no longer
- * satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
- * ordering may change the value of complex eigenvalues
- * (especially if the eigenvalue is ill-conditioned); in this
- * case INFO may be set to N+3 (see INFO below).
- *
- * SENSE (input) CHARACTER*1
- * Determines which reciprocal condition numbers are computed.
- * = 'N': None are computed;
- * = 'E': Computed for average of selected eigenvalues only;
- * = 'V': Computed for selected right invariant subspace only;
- * = 'B': Computed for both.
- * If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.
- *
- * N (input) INTEGER
- * The order of the matrix A. N >= 0.
- *
- * A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
- * On entry, the N-by-N matrix A.
- * On exit, A is overwritten by its real Schur form T.
- *
- * LDA (input) INTEGER
- * The leading dimension of the array A. LDA >= max(1,N).
- *
- * SDIM (output) INTEGER
- * If SORT = 'N', SDIM = 0.
- * If SORT = 'S', SDIM = number of eigenvalues (after sorting)
- * for which SELECT is true. (Complex conjugate
- * pairs for which SELECT is true for either
- * eigenvalue count as 2.)
- *
- * WR (output) DOUBLE PRECISION array, dimension (N)
- * WI (output) DOUBLE PRECISION array, dimension (N)
- * WR and WI contain the real and imaginary parts, respectively,
- * of the computed eigenvalues, in the same order that they
- * appear on the diagonal of the output Schur form T. Complex
- * conjugate pairs of eigenvalues appear consecutively with the
- * eigenvalue having the positive imaginary part first.
- *
- * VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
- * If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
- * vectors.
- * If JOBVS = 'N', VS is not referenced.
- *
- * LDVS (input) INTEGER
- * The leading dimension of the array VS. LDVS >= 1, and if
- * JOBVS = 'V', LDVS >= N.
- *
- * RCONDE (output) DOUBLE PRECISION
- * If SENSE = 'E' or 'B', RCONDE contains the reciprocal
- * condition number for the average of the selected eigenvalues.
- * Not referenced if SENSE = 'N' or 'V'.
- *
- * RCONDV (output) DOUBLE PRECISION
- * If SENSE = 'V' or 'B', RCONDV contains the reciprocal
- * condition number for the selected right invariant subspace.
- * Not referenced if SENSE = 'N' or 'E'.
- *
- * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
- * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
- *
- * LWORK (input) INTEGER
- * The dimension of the array WORK. LWORK >= max(1,3*N).
- * Also, if SENSE = 'E' or 'V' or 'B',
- * LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of
- * selected eigenvalues computed by this routine. Note that
- * N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only
- * returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or
- * 'B' this may not be large enough.
- * For good performance, LWORK must generally be larger.
- *
- * If LWORK = -1, then a workspace query is assumed; the routine
- * only calculates upper bounds on the optimal sizes of the
- * arrays WORK and IWORK, returns these values as the first
- * entries of the WORK and IWORK arrays, and no error messages
- * related to LWORK or LIWORK are issued by XERBLA.
- *
- * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
- * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
- *
- * LIWORK (input) INTEGER
- * The dimension of the array IWORK.
- * LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).
- * Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is
- * only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this
- * may not be large enough.
- *
- * If LIWORK = -1, then a workspace query is assumed; the
- * routine only calculates upper bounds on the optimal sizes of
- * the arrays WORK and IWORK, returns these values as the first
- * entries of the WORK and IWORK arrays, and no error messages
- * related to LWORK or LIWORK are issued by XERBLA.
- *
- * BWORK (workspace) LOGICAL array, dimension (N)
- * Not referenced if SORT = '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: the QR algorithm failed to compute all the
- * eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
- * contain those eigenvalues which have converged; if
- * JOBVS = 'V', VS contains the transformation which
- * reduces A to its partially converged Schur form.
- * = N+1: the eigenvalues could not be reordered because some
- * eigenvalues were too close to separate (the problem
- * is very ill-conditioned);
- * = N+2: after reordering, roundoff changed values of some
- * complex eigenvalues so that leading eigenvalues in
- * the Schur form no longer satisfy SELECT=.TRUE. This
- * could also be caused by underflow due to scaling.
- *
- * =====================================================================
- *
- * .. Parameters ..
- DOUBLE PRECISION ZERO, ONE
- PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
- * ..
- * .. Local Scalars ..
- LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB,
- $ WANTSE, WANTSN, WANTST, WANTSV, WANTVS
- INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL,
- $ IHI, ILO, INXT, IP, ITAU, IWRK, LIWRK, LWRK,
- $ MAXWRK, MINWRK
- DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM
- * ..
- * .. Local Arrays ..
- DOUBLE PRECISION DUM( 1 )
- * ..
- * .. External Subroutines ..
- EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY,
- $ DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA
- * ..
- * .. External Functions ..
- LOGICAL LSAME
- INTEGER ILAENV
- DOUBLE PRECISION DLAMCH, DLANGE
- EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE
- * ..
- * .. Intrinsic Functions ..
- INTRINSIC MAX, SQRT
- * ..
- * .. Executable Statements ..
- *
- * Test the input arguments
- *
- INFO = 0
- WANTVS = LSAME( JOBVS, 'V' )
- WANTST = LSAME( SORT, 'S' )
- WANTSN = LSAME( SENSE, 'N' )
- WANTSE = LSAME( SENSE, 'E' )
- WANTSV = LSAME( SENSE, 'V' )
- WANTSB = LSAME( SENSE, 'B' )
- LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
- IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN
- INFO = -1
- ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN
- INFO = -2
- ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR.
- $ ( .NOT.WANTST .AND. .NOT.WANTSN ) …
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