SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* Purpose
* =======
*
* DGETRF computes an LU factorization of a general M-by-N matrix A
* using partial pivoting with row interchanges.
*
* The factorization has the form
* A = P * L * U
* where P is a permutation matrix, L is lower triangular with unit
* diagonal elements (lower trapezoidal if m > n), and U is upper
* triangular (upper trapezoidal if m < n).
*
* This is the right-looking Level 3 BLAS version of the algorithm.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the M-by-N matrix to be factored.
* On exit, the factors L and U from the factorization
* A = P*L*U; the unit diagonal elements of L are not stored.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* IPIV (output) INTEGER array, dimension (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* 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 division by zero will occur if it is used
* to solve a system of equations.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IINFO, J, JB, NB
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DGETF2, DLASWP, DTRSM, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
EXTERNAL ILAENV
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment.
*
NB = ILAENV( 1, 'DGETRF', ' ', M, N, -1, -1 )
IF( NB.LE.1 .OR. NB.GE.MIN( M, N ) ) THEN
*
* Use unblocked code.
*
CALL DGETF2( M, N, A, LDA, IPIV, INFO )
ELSE
*
* Use blocked code.
*
DO 20 J = 1, MIN( M, N ), NB
JB = MIN( MIN( M, N )-J+1, NB )
*
* Factor diagonal and subdiagonal blocks and test for exact
* singularity.
*
CALL DGETF2( M-J+1, JB, A( J, J ), LDA, IPIV( J ), IINFO )
*
* Adjust INFO and the pivot indices.
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO + J - 1
DO 10 I = J, MIN( M, J+JB-1 )
IPIV( I ) = J - 1 + IPIV( I )
10 CONTINUE
*
* Apply interchanges to columns 1:J-1.
*
CALL DLASWP( J-1, A, LDA, J, J+JB-1, IPIV, 1 )
*
IF( J+JB.LE.N ) THEN
*
* Apply interchanges to columns J+JB:N.
*
CALL DLASWP( N-J-JB+1, A( 1, J+JB ), LDA, J, J+JB-1,
$ IPIV, 1 )
*
* Compute block row of U.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', JB,
$ N-J-JB+1, ONE, A( J, J ), LDA, A( J, J+JB ),
$ LDA )
IF( J+JB.LE.M ) THEN
*
* Update trailing submatrix.
*
CALL DGEMM( 'No transpose', 'No transpose', M-J-JB+1,
$ N-J-JB+1, JB, -ONE, A( J+JB, J ), LDA,
$ A( J, J+JB ), LDA, ONE, A( J+JB, J+JB ),
$ LDA )
END IF
END IF
20 CONTINUE
END IF
RETURN
*
* End of DGETRF
*
END
SUBROUTINE DGETF2( M, N, A, LDA, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* Purpose
* =======
*
* DGETF2 computes an LU factorization of a general m-by-n matrix A
* using partial pivoting with row interchanges.
*
* The factorization has the form
* A = P * L * U
* where P is a permutation matrix, L is lower triangular with unit
* diagonal elements (lower trapezoidal if m > n), and U is upper
* triangular (upper trapezoidal if m < n).
*
* This is the right-looking Level 2 BLAS version of the algorithm.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns of the matrix A. N >= 0.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the m by n matrix to be factored.
* On exit, the factors L and U from the factorization
* A = P*L*U; the unit diagonal elements of L are not stored.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* IPIV (output) INTEGER array, dimension (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
* > 0: if INFO = k, U(k,k) is exactly zero. The factorization
* has been completed, but the factor U is exactly
* singular, and division by zero will occur if it is used
* to solve a system of equations.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION SFMIN
INTEGER I, J, JP
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
INTEGER IDAMAX
EXTERNAL DLAMCH, IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -4
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Compute machine safe minimum
*
SFMIN = DLAMCH('S')
*
DO 10 J = 1, MIN( M, N )
*
* Find pivot and test for singularity.
*
JP = J - 1 + IDAMAX( M-J+1, A( J, J ), 1 )
IPIV( J ) = JP
IF( A( JP, J ).NE.ZERO ) THEN
*
* Apply the interchange to columns 1:N.
*
IF( JP.NE.J )
$ CALL DSWAP( N, A( J, 1 ), LDA, A( JP, 1 ), LDA )
*
* Compute elements J+1:M of J-th column.
*
IF( J.LT.M ) THEN
IF( ABS(A( J, J )) .GE. SFMIN ) THEN
CALL DSCAL( M-J, ONE / A( J, J ), A( J+1, J ), 1 )
ELSE
DO 20 I = 1, M-J
A( J+I, J ) = A( J+I, J ) / A( J, J )
20 CONTINUE
END IF
END IF
*
ELSE IF( INFO.EQ.0 ) THEN
*
INFO = J
END IF
*
IF( J.LT.MIN( M, N ) ) THEN
*
* Update trailing submatrix.
*
CALL DGER( M-J, N-J, -ONE, A( J+1, J ), 1, A( J, J+1 ), LDA,
$ A( J+1, J+1 ), LDA )
END IF
10 CONTINUE
RETURN
*
* End of DGETF2
*
END
SUBROUTINE DLASWP( N, A, LDA, K1, K2, IPIV, INCX )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INCX, K1, K2, LDA, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * )
* ..
*
* Purpose
* =======
*
* DLASWP performs a series of row interchanges on the matrix A.
* One row interchange is initiated for each of rows K1 through K2 of A.
*
* Arguments
* =========
*
* N (input) INTEGER
* The number of columns of the matrix A.
*
* A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
* On entry, the matrix of column dimension N to which the row
* interchanges will be applied.
* On exit, the permuted matrix.
*
* LDA (input) INTEGER
* The leading dimension of the array A.
*
* K1 (input) INTEGER
* The first element of IPIV for which a row interchange will
* be done.
*
* K2 (input) INTEGER
* The last element of IPIV for which a row interchange will
* be done.
*
* IPIV (input) INTEGER array, dimension (K2*abs(INCX))
* The vector of pivot indices. Only the elements in positions
* K1 through K2 of IPIV are accessed.
* IPIV(K) = L implies rows K and L are to be interchanged.
*
* INCX (input) INTEGER
* The increment between successive values of IPIV. If IPIV
* is negative, the pivots are applied in reverse order.
*
* Further Details
* ===============
*
* Modified by
* R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, I1, I2, INC, IP, IX, IX0, J, K, N32
DOUBLE PRECISION TEMP
* ..
* .. Executable Statements ..
*
* Interchange row I with row IPIV(I) for each of rows K1 through K2.
*
IF( INCX.GT.0 ) THEN
IX0 = K1
I1 = K1
I2 = K2
INC = 1
ELSE IF( INCX.LT.0 ) THEN
IX0 = 1 + ( 1-K2 )*INCX
I1 = K2
I2 = K1
INC = -1
ELSE
RETURN
END IF
*
N32 = ( N / 32 )*32
IF( N32.NE.0 ) THEN
DO 30 J = 1, N32, 32
IX = IX0
DO 20 I = I1, I2, INC
IP = IPIV( IX )
IF( IP.NE.I ) THEN
DO 10 K = J, J + 31
TEMP = A( I, K )
A( I, K ) = A( IP, K )
A( IP, K ) = TEMP
10 CONTINUE
END IF
IX = IX + INCX
20 CONTINUE
30 CONTINUE
END IF
IF( N32.NE.N ) THEN
N32 = N32 + 1
IX = IX0
DO 50 I = I1, I2, INC
IP = IPIV( IX )
IF( IP.NE.I ) THEN
DO 40 K = N32, N
TEMP = A( I, K )
A( I, K ) = A( IP, K )
A( IP, K ) = TEMP
40 CONTINUE
END IF
IX = IX + INCX
50 CONTINUE
END IF
*
RETURN
*
* End of DLASWP
*
END
SUBROUTINE DGETRS( TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* DGETRS solves a system of linear equations
* A * X = B or A' * X = B
* with a general N-by-N matrix A using the LU factorization computed
* by DGETRF.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations:
* = 'N': A * X = B (No transpose)
* = 'T': A'* X = B (Transpose)
* = 'C': A'* X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* 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 matrix B. NRHS >= 0.
*
* A (input) DOUBLE PRECISION array, dimension (LDA,N)
* The factors L and U from the factorization A = P*L*U
* as computed by DGETRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices from DGETRF; for 1<=i<=N, row i of the
* matrix was interchanged with row IPIV(i).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the 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
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLASWP, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( NRHS.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGETRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
IF( NOTRAN ) THEN
*
* Solve A * X = B.
*
* Apply row interchanges to the right hand sides.
*
CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, 1 )
*
* Solve L*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit', N, NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve U*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, ONE, A, LDA, B, LDB )
ELSE
*
* Solve A' * X = B.
*
* Solve U'*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', N, NRHS,
$ ONE, A, LDA, B, LDB )
*
* Solve L'*X = B, overwriting B with X.
*
CALL DTRSM( 'Left', 'Lower', 'Transpose', 'Unit', N, NRHS, ONE,
$ A, LDA, B, LDB )
*
* Apply row interchanges to the solution vectors.
*
CALL DLASWP( NRHS, B, LDB, 1, N, IPIV, -1 )
END IF
*
RETURN
*
* End of DGETRS
*
END
SUBROUTINE DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* Purpose
* =======
*
* DGBTF2 computes an LU factorization of a real m-by-n band matrix A
* using partial pivoting with row interchanges.
*
* This is the unblocked version of the algorithm, calling Level 2 BLAS.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns 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.
*
* 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(m,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 (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* 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 division by zero will occur if it is used
* to solve a system of equations.
*
* 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.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, JP, JU, KM, KV
* ..
* .. External Functions ..
INTEGER IDAMAX
EXTERNAL IDAMAX
* ..
* .. External Subroutines ..
EXTERNAL DGER, DSCAL, DSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* KV is the number of superdiagonals in the factor U, allowing for
* fill-in.
*
KV = KU + KL
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KV+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTF2', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Gaussian elimination with partial pivoting
*
* Set fill-in elements in columns KU+2 to KV to zero.
*
DO 20 J = KU + 2, MIN( KV, N )
DO 10 I = KV - J + 2, KL
AB( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* JU is the index of the last column affected by the current stage
* of the factorization.
*
JU = 1
*
DO 40 J = 1, MIN( M, N )
*
* Set fill-in elements in column J+KV to zero.
*
IF( J+KV.LE.N ) THEN
DO 30 I = 1, KL
AB( I, J+KV ) = ZERO
30 CONTINUE
END IF
*
* Find pivot and test for singularity. KM is the number of
* subdiagonal elements in the current column.
*
KM = MIN( KL, M-J )
JP = IDAMAX( KM+1, AB( KV+1, J ), 1 )
IPIV( J ) = JP + J - 1
IF( AB( KV+JP, J ).NE.ZERO ) THEN
JU = MAX( JU, MIN( J+KU+JP-1, N ) )
*
* Apply interchange to columns J to JU.
*
IF( JP.NE.1 )
$ CALL DSWAP( JU-J+1, AB( KV+JP, J ), LDAB-1,
$ AB( KV+1, J ), LDAB-1 )
*
IF( KM.GT.0 ) THEN
*
* Compute multipliers.
*
CALL DSCAL( KM, ONE / AB( KV+1, J ), AB( KV+2, J ), 1 )
*
* Update trailing submatrix within the band.
*
IF( JU.GT.J )
$ CALL DGER( KM, JU-J, -ONE, AB( KV+2, J ), 1,
$ AB( KV, J+1 ), LDAB-1, AB( KV+1, J+1 ),
$ LDAB-1 )
END IF
ELSE
*
* If pivot is zero, set INFO to the index of the pivot
* unless a zero pivot has already been found.
*
IF( INFO.EQ.0 )
$ INFO = J
END IF
40 CONTINUE
RETURN
*
* End of DGBTF2
*
END
SUBROUTINE DGBTRF( M, N, KL, KU, AB, LDAB, IPIV, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO, KL, KU, LDAB, M, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * )
* ..
*
* Purpose
* =======
*
* DGBTRF computes an LU factorization of a real m-by-n band matrix A
* using partial pivoting with row interchanges.
*
* This is the blocked version of the algorithm, calling Level 3 BLAS.
*
* Arguments
* =========
*
* M (input) INTEGER
* The number of rows of the matrix A. M >= 0.
*
* N (input) INTEGER
* The number of columns 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.
*
* 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(m,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 (min(M,N))
* The pivot indices; for 1 <= i <= min(M,N), row i of the
* matrix was interchanged with row IPIV(i).
*
* 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 division by zero will occur if it is used
* to solve a system of equations.
*
* 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.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
INTEGER NBMAX, LDWORK
PARAMETER ( NBMAX = 64, LDWORK = NBMAX+1 )
* ..
* .. Local Scalars ..
INTEGER I, I2, I3, II, IP, J, J2, J3, JB, JJ, JM, JP,
$ JU, K2, KM, KV, NB, NW
DOUBLE PRECISION TEMP
* ..
* .. Local Arrays ..
DOUBLE PRECISION WORK13( LDWORK, NBMAX ),
$ WORK31( LDWORK, NBMAX )
* ..
* .. External Functions ..
INTEGER IDAMAX, ILAENV
EXTERNAL IDAMAX, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL DCOPY, DGBTF2, DGEMM, DGER, DLASWP, DSCAL,
$ DSWAP, DTRSM, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* KV is the number of superdiagonals in the factor U, allowing for
* fill-in
*
KV = KU + KL
*
* Test the input parameters.
*
INFO = 0
IF( M.LT.0 ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( LDAB.LT.KL+KV+1 ) THEN
INFO = -6
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTRF', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( M.EQ.0 .OR. N.EQ.0 )
$ RETURN
*
* Determine the block size for this environment
*
NB = ILAENV( 1, 'DGBTRF', ' ', M, N, KL, KU )
*
* The block size must not exceed the limit set by the size of the
* local arrays WORK13 and WORK31.
*
NB = MIN( NB, NBMAX )
*
IF( NB.LE.1 .OR. NB.GT.KL ) THEN
*
* Use unblocked code
*
CALL DGBTF2( M, N, KL, KU, AB, LDAB, IPIV, INFO )
ELSE
*
* Use blocked code
*
* Zero the superdiagonal elements of the work array WORK13
*
DO 20 J = 1, NB
DO 10 I = 1, J - 1
WORK13( I, J ) = ZERO
10 CONTINUE
20 CONTINUE
*
* Zero the subdiagonal elements of the work array WORK31
*
DO 40 J = 1, NB
DO 30 I = J + 1, NB
WORK31( I, J ) = ZERO
30 CONTINUE
40 CONTINUE
*
* Gaussian elimination with partial pivoting
*
* Set fill-in elements in columns KU+2 to KV to zero
*
DO 60 J = KU + 2, MIN( KV, N )
DO 50 I = KV - J + 2, KL
AB( I, J ) = ZERO
50 CONTINUE
60 CONTINUE
*
* JU is the index of the last column affected by the current
* stage of the factorization
*
JU = 1
*
DO 180 J = 1, MIN( M, N ), NB
JB = MIN( NB, MIN( M, N )-J+1 )
*
* The active part of the matrix is partitioned
*
* A11 A12 A13
* A21 A22 A23
* A31 A32 A33
*
* Here A11, A21 and A31 denote the current block of JB columns
* which is about to be factorized. The number of rows in the
* partitioning are JB, I2, I3 respectively, and the numbers
* of columns are JB, J2, J3. The superdiagonal elements of A13
* and the subdiagonal elements of A31 lie outside the band.
*
I2 = MIN( KL-JB, M-J-JB+1 )
I3 = MIN( JB, M-J-KL+1 )
*
* J2 and J3 are computed after JU has been updated.
*
* Factorize the current block of JB columns
*
DO 80 JJ = J, J + JB - 1
*
* Set fill-in elements in column JJ+KV to zero
*
IF( JJ+KV.LE.N ) THEN
DO 70 I = 1, KL
AB( I, JJ+KV ) = ZERO
70 CONTINUE
END IF
*
* Find pivot and test for singularity. KM is the number of
* subdiagonal elements in the current column.
*
KM = MIN( KL, M-JJ )
JP = IDAMAX( KM+1, AB( KV+1, JJ ), 1 )
IPIV( JJ ) = JP + JJ - J
IF( AB( KV+JP, JJ ).NE.ZERO ) THEN
JU = MAX( JU, MIN( JJ+KU+JP-1, N ) )
IF( JP.NE.1 ) THEN
*
* Apply interchange to columns J to J+JB-1
*
IF( JP+JJ-1.LT.J+KL ) THEN
*
CALL DSWAP( JB, AB( KV+1+JJ-J, J ), LDAB-1,
$ AB( KV+JP+JJ-J, J ), LDAB-1 )
ELSE
*
* The interchange affects columns J to JJ-1 of A31
* which are stored in the work array WORK31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
CALL DSWAP( J+JB-JJ, AB( KV+1, JJ ), LDAB-1,
$ AB( KV+JP, JJ ), LDAB-1 )
END IF
END IF
*
* Compute multipliers
*
CALL DSCAL( KM, ONE / AB( KV+1, JJ ), AB( KV+2, JJ ),
$ 1 )
*
* Update trailing submatrix within the band and within
* the current block. JM is the index of the last column
* which needs to be updated.
*
JM = MIN( JU, J+JB-1 )
IF( JM.GT.JJ )
$ CALL DGER( KM, JM-JJ, -ONE, AB( KV+2, JJ ), 1,
$ AB( KV, JJ+1 ), LDAB-1,
$ AB( KV+1, JJ+1 ), LDAB-1 )
ELSE
*
* If pivot is zero, set INFO to the index of the pivot
* unless a zero pivot has already been found.
*
IF( INFO.EQ.0 )
$ INFO = JJ
END IF
*
* Copy current column of A31 into the work array WORK31
*
NW = MIN( JJ-J+1, I3 )
IF( NW.GT.0 )
$ CALL DCOPY( NW, AB( KV+KL+1-JJ+J, JJ ), 1,
$ WORK31( 1, JJ-J+1 ), 1 )
80 CONTINUE
IF( J+JB.LE.N ) THEN
*
* Apply the row interchanges to the other blocks.
*
J2 = MIN( JU-J+1, KV ) - JB
J3 = MAX( 0, JU-J-KV+1 )
*
* Use DLASWP to apply the row interchanges to A12, A22, and
* A32.
*
CALL DLASWP( J2, AB( KV+1-JB, J+JB ), LDAB-1, 1, JB,
$ IPIV( J ), 1 )
*
* Adjust the pivot indices.
*
DO 90 I = J, J + JB - 1
IPIV( I ) = IPIV( I ) + J - 1
90 CONTINUE
*
* Apply the row interchanges to A13, A23, and A33
* columnwise.
*
K2 = J - 1 + JB + J2
DO 110 I = 1, J3
JJ = K2 + I
DO 100 II = J + I - 1, J + JB - 1
IP = IPIV( II )
IF( IP.NE.II ) THEN
TEMP = AB( KV+1+II-JJ, JJ )
AB( KV+1+II-JJ, JJ ) = AB( KV+1+IP-JJ, JJ )
AB( KV+1+IP-JJ, JJ ) = TEMP
END IF
100 CONTINUE
110 CONTINUE
*
* Update the relevant part of the trailing submatrix
*
IF( J2.GT.0 ) THEN
*
* Update A12
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
$ JB, J2, ONE, AB( KV+1, J ), LDAB-1,
$ AB( KV+1-JB, J+JB ), LDAB-1 )
*
IF( I2.GT.0 ) THEN
*
* Update A22
*
CALL DGEMM( 'No transpose', 'No transpose', I2, J2,
$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
$ AB( KV+1, J+JB ), LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Update A32
*
CALL DGEMM( 'No transpose', 'No transpose', I3, J2,
$ JB, -ONE, WORK31, LDWORK,
$ AB( KV+1-JB, J+JB ), LDAB-1, ONE,
$ AB( KV+KL+1-JB, J+JB ), LDAB-1 )
END IF
END IF
*
IF( J3.GT.0 ) THEN
*
* Copy the lower triangle of A13 into the work array
* WORK13
*
DO 130 JJ = 1, J3
DO 120 II = JJ, JB
WORK13( II, JJ ) = AB( II-JJ+1, JJ+J+KV-1 )
120 CONTINUE
130 CONTINUE
*
* Update A13 in the work array
*
CALL DTRSM( 'Left', 'Lower', 'No transpose', 'Unit',
$ JB, J3, ONE, AB( KV+1, J ), LDAB-1,
$ WORK13, LDWORK )
*
IF( I2.GT.0 ) THEN
*
* Update A23
*
CALL DGEMM( 'No transpose', 'No transpose', I2, J3,
$ JB, -ONE, AB( KV+1+JB, J ), LDAB-1,
$ WORK13, LDWORK, ONE, AB( 1+JB, J+KV ),
$ LDAB-1 )
END IF
*
IF( I3.GT.0 ) THEN
*
* Update A33
*
CALL DGEMM( 'No transpose', 'No transpose', I3, J3,
$ JB, -ONE, WORK31, LDWORK, WORK13,
$ LDWORK, ONE, AB( 1+KL, J+KV ), LDAB-1 )
END IF
*
* Copy the lower triangle of A13 back into place
*
DO 150 JJ = 1, J3
DO 140 II = JJ, JB
AB( II-JJ+1, JJ+J+KV-1 ) = WORK13( II, JJ )
140 CONTINUE
150 CONTINUE
END IF
ELSE
*
* Adjust the pivot indices.
*
DO 160 I = J, J + JB - 1
IPIV( I ) = IPIV( I ) + J - 1
160 CONTINUE
END IF
*
* Partially undo the interchanges in the current block to
* restore the upper triangular form of A31 and copy the upper
* triangle of A31 back into place
*
DO 170 JJ = J + JB - 1, J, -1
JP = IPIV( JJ ) - JJ + 1
IF( JP.NE.1 ) THEN
*
* Apply interchange to columns J to JJ-1
*
IF( JP+JJ-1.LT.J+KL ) THEN
*
* The interchange does not affect A31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ AB( KV+JP+JJ-J, J ), LDAB-1 )
ELSE
*
* The interchange does affect A31
*
CALL DSWAP( JJ-J, AB( KV+1+JJ-J, J ), LDAB-1,
$ WORK31( JP+JJ-J-KL, 1 ), LDWORK )
END IF
END IF
*
* Copy the current column of A31 back into place
*
NW = MIN( I3, JJ-J+1 )
IF( NW.GT.0 )
$ CALL DCOPY( NW, WORK31( 1, JJ-J+1 ), 1,
$ AB( KV+KL+1-JJ+J, JJ ), 1 )
170 CONTINUE
180 CONTINUE
END IF
*
RETURN
*
* End of DGBTRF
*
END
SUBROUTINE DGBTRS( TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
$ INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, KL, KU, LDAB, LDB, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION AB( LDAB, * ), B( LDB, * )
* ..
*
* Purpose
* =======
*
* DGBTRS solves a system of linear equations
* A * X = B or A' * X = B
* with a general band matrix A using the LU factorization computed
* by DGBTRF.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER*1
* Specifies the form of the system of equations.
* = 'N': A * X = B (No transpose)
* = 'T': A'* X = B (Transpose)
* = 'C': A'* X = B (Conjugate transpose = Transpose)
*
* N (input) INTEGER
* 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) DOUBLE PRECISION array, dimension (LDAB,N)
* 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.
*
* LDAB (input) INTEGER
* The leading dimension of the array AB. LDAB >= 2*KL+KU+1.
*
* IPIV (input) INTEGER array, dimension (N)
* The pivot indices; for 1 <= i <= N, row i of the matrix was
* interchanged with row IPIV(i).
*
* B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
* On entry, the right hand side matrix B.
* On exit, the 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
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LNOTI, NOTRAN
INTEGER I, J, KD, L, LM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DGER, DSWAP, DTBSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOTRAN = LSAME( TRANS, 'N' )
IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( KL.LT.0 ) THEN
INFO = -3
ELSE IF( KU.LT.0 ) THEN
INFO = -4
ELSE IF( NRHS.LT.0 ) THEN
INFO = -5
ELSE IF( LDAB.LT.( 2*KL+KU+1 ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGBTRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 .OR. NRHS.EQ.0 )
$ RETURN
*
KD = KU + KL + 1
LNOTI = KL.GT.0
*
IF( NOTRAN ) THEN
*
* Solve A*X = B.
*
* Solve L*X = B, overwriting B with X.
*
* L is represented as a product of permutations and unit lower
* triangular matrices L = P(1) * L(1) * ... * P(n-1) * L(n-1),
* where each transformation L(i) is a rank-one modification of
* the identity matrix.
*
IF( LNOTI ) THEN
DO 10 J = 1, N - 1
LM = MIN( KL, N-J )
L = IPIV( J )
IF( L.NE.J )
$ CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
CALL DGER( LM, NRHS, -ONE, AB( KD+1, J ), 1, B( J, 1 ),
$ LDB, B( J+1, 1 ), LDB )
10 CONTINUE
END IF
*
DO 20 I = 1, NRHS
*
* Solve U*X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'No transpose', 'Non-unit', N, KL+KU,
$ AB, LDAB, B( 1, I ), 1 )
20 CONTINUE
*
ELSE
*
* Solve A'*X = B.
*
DO 30 I = 1, NRHS
*
* Solve U'*X = B, overwriting B with X.
*
CALL DTBSV( 'Upper', 'Transpose', 'Non-unit', N, KL+KU, AB,
$ LDAB, B( 1, I ), 1 )
30 CONTINUE
*
* Solve L'*X = B, overwriting B with X.
*
IF( LNOTI ) THEN
DO 40 J = N - 1, 1, -1
LM = MIN( KL, N-J )
CALL DGEMV( 'Transpose', LM, NRHS, -ONE, B( J+1, 1 ),
$ LDB, AB( KD+1, J ), 1, ONE, B( J, 1 ), LDB )
L = IPIV( J )
IF( L.NE.J )
$ CALL DSWAP( NRHS, B( L, 1 ), LDB, B( J, 1 ), LDB )
40 CONTINUE
END IF
END IF
RETURN
*
* End of DGBTRS
*
END
SUBROUTINE DGER(M,N,ALPHA,X,INCX,Y,INCY,A,LDA)
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA
INTEGER INCX,INCY,LDA,M,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
* Purpose
* =======
*
* DGER performs the rank 1 operation
*
* A := alpha*x*y' + A,
*
* where alpha is a scalar, x is an m element vector, y is an n element
* vector and A is an m by n matrix.
*
* Arguments
* ==========
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix A.
* M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - DOUBLE PRECISION.
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* X - DOUBLE PRECISION array of dimension at least
* ( 1 + ( m - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the m
* element vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* Y - DOUBLE PRECISION array of dimension at least
* ( 1 + ( n - 1 )*abs( INCY ) ).
* Before entry, the incremented array Y must contain the n
* element vector y.
* Unchanged on exit.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
* Before entry, the leading m by n part of the array A must
* contain the matrix of coefficients. On exit, A is
* overwritten by the updated matrix.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,IX,J,JY,KX
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
*
* Test the input parameters.
*
INFO = 0
IF (M.LT.0) THEN
INFO = 1
ELSE IF (N.LT.0) THEN
INFO = 2
ELSE IF (INCX.EQ.0) THEN
INFO = 5
ELSE IF (INCY.EQ.0) THEN
INFO = 7
ELSE IF (LDA.LT.MAX(1,M)) THEN
INFO = 9
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DGER ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((M.EQ.0) .OR. (N.EQ.0) .OR. (ALPHA.EQ.ZERO)) RETURN
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through A.
*
IF (INCY.GT.0) THEN
JY = 1
ELSE
JY = 1 - (N-1)*INCY
END IF
IF (INCX.EQ.1) THEN
DO 20 J = 1,N
IF (Y(JY).NE.ZERO) THEN
TEMP = ALPHA*Y(JY)
DO 10 I = 1,M
A(I,J) = A(I,J) + X(I)*TEMP
10 CONTINUE
END IF
JY = JY + INCY
20 CONTINUE
ELSE
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (M-1)*INCX
END IF
DO 40 J = 1,N
IF (Y(JY).NE.ZERO) THEN
TEMP = ALPHA*Y(JY)
IX = KX
DO 30 I = 1,M
A(I,J) = A(I,J) + X(IX)*TEMP
IX = IX + INCX
30 CONTINUE
END IF
JY = JY + INCY
40 CONTINUE
END IF
*
RETURN
*
* End of DGER .
*
END
SUBROUTINE DSWAP(N,DX,INCX,DY,INCY)
* .. Scalar Arguments ..
INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*),DY(*)
* ..
*
* Purpose
* =======
*
* interchanges two vectors.
* uses unrolled loops for increments equal one.
* jack dongarra, linpack, 3/11/78.
* modified 12/3/93, array(1) declarations changed to array(*)
*
*
* .. Local Scalars ..
DOUBLE PRECISION DTEMP
INTEGER I,IX,IY,M,MP1
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
IF (N.LE.0) RETURN
IF (INCX.EQ.1 .AND. INCY.EQ.1) GO TO 20
*
* code for unequal increments or equal increments not equal
* to 1
*
IX = 1
IY = 1
IF (INCX.LT.0) IX = (-N+1)*INCX + 1
IF (INCY.LT.0) IY = (-N+1)*INCY + 1
DO 10 I = 1,N
DTEMP = DX(IX)
DX(IX) = DY(IY)
DY(IY) = DTEMP
IX = IX + INCX
IY = IY + INCY
10 CONTINUE
RETURN
*
* code for both increments equal to 1
*
*
* clean-up loop
*
20 M = MOD(N,3)
IF (M.EQ.0) GO TO 40
DO 30 I = 1,M
DTEMP = DX(I)
DX(I) = DY(I)
DY(I) = DTEMP
30 CONTINUE
IF (N.LT.3) RETURN
40 MP1 = M + 1
DO 50 I = MP1,N,3
DTEMP = DX(I)
DX(I) = DY(I)
DY(I) = DTEMP
DTEMP = DX(I+1)
DX(I+1) = DY(I+1)
DY(I+1) = DTEMP
DTEMP = DX(I+2)
DX(I+2) = DY(I+2)
DY(I+2) = DTEMP
50 CONTINUE
RETURN
END
SUBROUTINE XERBLA(SRNAME,INFO)
*
* -- LAPACK auxiliary routine (preliminary version) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER INFO
CHARACTER*6 SRNAME
* ..
*
* Purpose
* =======
*
* XERBLA is an error handler for the LAPACK routines.
* It is called by an LAPACK routine if an input parameter has an
* invalid value. A message is printed and execution stops.
*
* Installers may consider modifying the STOP statement in order to
* call system-specific exception-handling facilities.
*
* Arguments
* =========
*
* SRNAME (input) CHARACTER*6
* The name of the routine which called XERBLA.
*
* INFO (input) INTEGER
* The position of the invalid parameter in the parameter list
* of the calling routine.
*
*
WRITE (*,FMT=9999) SRNAME,INFO
*
STOP
*
9999 FORMAT (' ** On entry to ',A6,' parameter number ',I2,' had ',
+ 'an illegal value')
*
* End of XERBLA
*
END
SUBROUTINE DTRSM(SIDE,UPLO,TRANSA,DIAG,M,N,ALPHA,A,LDA,B,LDB)
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA
INTEGER LDA,LDB,M,N
CHARACTER DIAG,SIDE,TRANSA,UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),B(LDB,*)
* ..
*
* Purpose
* =======
*
* DTRSM solves one of the matrix equations
*
* op( A )*X = alpha*B, or X*op( A ) = alpha*B,
*
* where alpha is a scalar, X and B are m by n matrices, A is a unit, or
* non-unit, upper or lower triangular matrix and op( A ) is one of
*
* op( A ) = A or op( A ) = A'.
*
* The matrix X is overwritten on B.
*
* Arguments
* ==========
*
* SIDE - CHARACTER*1.
* On entry, SIDE specifies whether op( A ) appears on the left
* or right of X as follows:
*
* SIDE = 'L' or 'l' op( A )*X = alpha*B.
*
* SIDE = 'R' or 'r' X*op( A ) = alpha*B.
*
* Unchanged on exit.
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the matrix A is an upper or
* lower triangular matrix as follows:
*
* UPLO = 'U' or 'u' A is an upper triangular matrix.
*
* UPLO = 'L' or 'l' A is a lower triangular matrix.
*
* Unchanged on exit.
*
* TRANSA - CHARACTER*1.
* On entry, TRANSA specifies the form of op( A ) to be used in
* the matrix multiplication as follows:
*
* TRANSA = 'N' or 'n' op( A ) = A.
*
* TRANSA = 'T' or 't' op( A ) = A'.
*
* TRANSA = 'C' or 'c' op( A ) = A'.
*
* Unchanged on exit.
*
* DIAG - CHARACTER*1.
* On entry, DIAG specifies whether or not A is unit triangular
* as follows:
*
* DIAG = 'U' or 'u' A is assumed to be unit triangular.
*
* DIAG = 'N' or 'n' A is not assumed to be unit
* triangular.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of B. M must be at
* least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of B. N must be
* at least zero.
* Unchanged on exit.
*
* ALPHA - DOUBLE PRECISION.
* On entry, ALPHA specifies the scalar alpha. When alpha is
* zero then A is not referenced and B need not be set before
* entry.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of DIMENSION ( LDA, k ), where k is m
* when SIDE = 'L' or 'l' and is n when SIDE = 'R' or 'r'.
* Before entry with UPLO = 'U' or 'u', the leading k by k
* upper triangular part of the array A must contain the upper
* triangular matrix and the strictly lower triangular part of
* A is not referenced.
* Before entry with UPLO = 'L' or 'l', the leading k by k
* lower triangular part of the array A must contain the lower
* triangular matrix and the strictly upper triangular part of
* A is not referenced.
* Note that when DIAG = 'U' or 'u', the diagonal elements of
* A are not referenced either, but are assumed to be unity.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When SIDE = 'L' or 'l' then
* LDA must be at least max( 1, m ), when SIDE = 'R' or 'r'
* then LDA must be at least max( 1, n ).
* Unchanged on exit.
*
* B - DOUBLE PRECISION array of DIMENSION ( LDB, n ).
* Before entry, the leading m by n part of the array B must
* contain the right-hand side matrix B, and on exit is
* overwritten by the solution matrix X.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. LDB must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,J,K,NROWA
LOGICAL LSIDE,NOUNIT,UPPER
* ..
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
*
* Test the input parameters.
*
LSIDE = LSAME(SIDE,'L')
IF (LSIDE) THEN
NROWA = M
ELSE
NROWA = N
END IF
NOUNIT = LSAME(DIAG,'N')
UPPER = LSAME(UPLO,'U')
*
INFO = 0
IF ((.NOT.LSIDE) .AND. (.NOT.LSAME(SIDE,'R'))) THEN
INFO = 1
ELSE IF ((.NOT.UPPER) .AND. (.NOT.LSAME(UPLO,'L'))) THEN
INFO = 2
ELSE IF ((.NOT.LSAME(TRANSA,'N')) .AND.
+ (.NOT.LSAME(TRANSA,'T')) .AND.
+ (.NOT.LSAME(TRANSA,'C'))) THEN
INFO = 3
ELSE IF ((.NOT.LSAME(DIAG,'U')) .AND. (.NOT.LSAME(DIAG,'N'))) THEN
INFO = 4
ELSE IF (M.LT.0) THEN
INFO = 5
ELSE IF (N.LT.0) THEN
INFO = 6
ELSE IF (LDA.LT.MAX(1,NROWA)) THEN
INFO = 9
ELSE IF (LDB.LT.MAX(1,M)) THEN
INFO = 11
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DTRSM ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF (M.EQ.0 .OR. N.EQ.0) RETURN
*
* And when alpha.eq.zero.
*
IF (ALPHA.EQ.ZERO) THEN
DO 20 J = 1,N
DO 10 I = 1,M
B(I,J) = ZERO
10 CONTINUE
20 CONTINUE
RETURN
END IF
*
* Start the operations.
*
IF (LSIDE) THEN
IF (LSAME(TRANSA,'N')) THEN
*
* Form B := alpha*inv( A )*B.
*
IF (UPPER) THEN
DO 60 J = 1,N
IF (ALPHA.NE.ONE) THEN
DO 30 I = 1,M
B(I,J) = ALPHA*B(I,J)
30 CONTINUE
END IF
DO 50 K = M,1,-1
IF (B(K,J).NE.ZERO) THEN
IF (NOUNIT) B(K,J) = B(K,J)/A(K,K)
DO 40 I = 1,K - 1
B(I,J) = B(I,J) - B(K,J)*A(I,K)
40 CONTINUE
END IF
50 CONTINUE
60 CONTINUE
ELSE
DO 100 J = 1,N
IF (ALPHA.NE.ONE) THEN
DO 70 I = 1,M
B(I,J) = ALPHA*B(I,J)
70 CONTINUE
END IF
DO 90 K = 1,M
IF (B(K,J).NE.ZERO) THEN
IF (NOUNIT) B(K,J) = B(K,J)/A(K,K)
DO 80 I = K + 1,M
B(I,J) = B(I,J) - B(K,J)*A(I,K)
80 CONTINUE
END IF
90 CONTINUE
100 CONTINUE
END IF
ELSE
*
* Form B := alpha*inv( A' )*B.
*
IF (UPPER) THEN
DO 130 J = 1,N
DO 120 I = 1,M
TEMP = ALPHA*B(I,J)
DO 110 K = 1,I - 1
TEMP = TEMP - A(K,I)*B(K,J)
110 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(I,I)
B(I,J) = TEMP
120 CONTINUE
130 CONTINUE
ELSE
DO 160 J = 1,N
DO 150 I = M,1,-1
TEMP = ALPHA*B(I,J)
DO 140 K = I + 1,M
TEMP = TEMP - A(K,I)*B(K,J)
140 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(I,I)
B(I,J) = TEMP
150 CONTINUE
160 CONTINUE
END IF
END IF
ELSE
IF (LSAME(TRANSA,'N')) THEN
*
* Form B := alpha*B*inv( A ).
*
IF (UPPER) THEN
DO 210 J = 1,N
IF (ALPHA.NE.ONE) THEN
DO 170 I = 1,M
B(I,J) = ALPHA*B(I,J)
170 CONTINUE
END IF
DO 190 K = 1,J - 1
IF (A(K,J).NE.ZERO) THEN
DO 180 I = 1,M
B(I,J) = B(I,J) - A(K,J)*B(I,K)
180 CONTINUE
END IF
190 CONTINUE
IF (NOUNIT) THEN
TEMP = ONE/A(J,J)
DO 200 I = 1,M
B(I,J) = TEMP*B(I,J)
200 CONTINUE
END IF
210 CONTINUE
ELSE
DO 260 J = N,1,-1
IF (ALPHA.NE.ONE) THEN
DO 220 I = 1,M
B(I,J) = ALPHA*B(I,J)
220 CONTINUE
END IF
DO 240 K = J + 1,N
IF (A(K,J).NE.ZERO) THEN
DO 230 I = 1,M
B(I,J) = B(I,J) - A(K,J)*B(I,K)
230 CONTINUE
END IF
240 CONTINUE
IF (NOUNIT) THEN
TEMP = ONE/A(J,J)
DO 250 I = 1,M
B(I,J) = TEMP*B(I,J)
250 CONTINUE
END IF
260 CONTINUE
END IF
ELSE
*
* Form B := alpha*B*inv( A' ).
*
IF (UPPER) THEN
DO 310 K = N,1,-1
IF (NOUNIT) THEN
TEMP = ONE/A(K,K)
DO 270 I = 1,M
B(I,K) = TEMP*B(I,K)
270 CONTINUE
END IF
DO 290 J = 1,K - 1
IF (A(J,K).NE.ZERO) THEN
TEMP = A(J,K)
DO 280 I = 1,M
B(I,J) = B(I,J) - TEMP*B(I,K)
280 CONTINUE
END IF
290 CONTINUE
IF (ALPHA.NE.ONE) THEN
DO 300 I = 1,M
B(I,K) = ALPHA*B(I,K)
300 CONTINUE
END IF
310 CONTINUE
ELSE
DO 360 K = 1,N
IF (NOUNIT) THEN
TEMP = ONE/A(K,K)
DO 320 I = 1,M
B(I,K) = TEMP*B(I,K)
320 CONTINUE
END IF
DO 340 J = K + 1,N
IF (A(J,K).NE.ZERO) THEN
TEMP = A(J,K)
DO 330 I = 1,M
B(I,J) = B(I,J) - TEMP*B(I,K)
330 CONTINUE
END IF
340 CONTINUE
IF (ALPHA.NE.ONE) THEN
DO 350 I = 1,M
B(I,K) = ALPHA*B(I,K)
350 CONTINUE
END IF
360 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of DTRSM .
*
END
SUBROUTINE DSCAL(N,DA,DX,INCX)
* .. Scalar Arguments ..
DOUBLE PRECISION DA
INTEGER INCX,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*)
* ..
*
* Purpose
* =======
**
* scales a vector by a constant.
* uses unrolled loops for increment equal to one.
* jack dongarra, linpack, 3/11/78.
* modified 3/93 to return if incx .le. 0.
* modified 12/3/93, array(1) declarations changed to array(*)
*
*
* .. Local Scalars ..
INTEGER I,M,MP1,NINCX
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
IF (N.LE.0 .OR. INCX.LE.0) RETURN
IF (INCX.EQ.1) GO TO 20
*
* code for increment not equal to 1
*
NINCX = N*INCX
DO 10 I = 1,NINCX,INCX
DX(I) = DA*DX(I)
10 CONTINUE
RETURN
*
* code for increment equal to 1
*
*
* clean-up loop
*
20 M = MOD(N,5)
IF (M.EQ.0) GO TO 40
DO 30 I = 1,M
DX(I) = DA*DX(I)
30 CONTINUE
IF (N.LT.5) RETURN
40 MP1 = M + 1
DO 50 I = MP1,N,5
DX(I) = DA*DX(I)
DX(I+1) = DA*DX(I+1)
DX(I+2) = DA*DX(I+2)
DX(I+3) = DA*DX(I+3)
DX(I+4) = DA*DX(I+4)
50 CONTINUE
RETURN
END
SUBROUTINE DCOPY(N,DX,INCX,DY,INCY)
* .. Scalar Arguments ..
INTEGER INCX,INCY,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*),DY(*)
* ..
*
* Purpose
* =======
*
* copies a vector, x, to a vector, y.
* uses unrolled loops for increments equal to one.
* jack dongarra, linpack, 3/11/78.
* modified 12/3/93, array(1) declarations changed to array(*)
*
*
* .. Local Scalars ..
INTEGER I,IX,IY,M,MP1
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
IF (N.LE.0) RETURN
IF (INCX.EQ.1 .AND. INCY.EQ.1) GO TO 20
*
* code for unequal increments or equal increments
* not equal to 1
*
IX = 1
IY = 1
IF (INCX.LT.0) IX = (-N+1)*INCX + 1
IF (INCY.LT.0) IY = (-N+1)*INCY + 1
DO 10 I = 1,N
DY(IY) = DX(IX)
IX = IX + INCX
IY = IY + INCY
10 CONTINUE
RETURN
*
* code for both increments equal to 1
*
*
* clean-up loop
*
20 M = MOD(N,7)
IF (M.EQ.0) GO TO 40
DO 30 I = 1,M
DY(I) = DX(I)
30 CONTINUE
IF (N.LT.7) RETURN
40 MP1 = M + 1
DO 50 I = MP1,N,7
DY(I) = DX(I)
DY(I+1) = DX(I+1)
DY(I+2) = DX(I+2)
DY(I+3) = DX(I+3)
DY(I+4) = DX(I+4)
DY(I+5) = DX(I+5)
DY(I+6) = DX(I+6)
50 CONTINUE
RETURN
END
SUBROUTINE DTBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX)
* .. Scalar Arguments ..
INTEGER INCX,K,LDA,N
CHARACTER DIAG,TRANS,UPLO
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*)
* ..
*
* Purpose
* =======
*
* DTBSV solves one of the systems of equations
*
* A*x = b, or A'*x = b,
*
* where b and x are n element vectors and A is an n by n unit, or
* non-unit, upper or lower triangular band matrix, with ( k + 1 )
* diagonals.
*
* No test for singularity or near-singularity is included in this
* routine. Such tests must be performed before calling this routine.
*
* Arguments
* ==========
*
* UPLO - CHARACTER*1.
* On entry, UPLO specifies whether the matrix is an upper or
* lower triangular matrix as follows:
*
* UPLO = 'U' or 'u' A is an upper triangular matrix.
*
* UPLO = 'L' or 'l' A is a lower triangular matrix.
*
* Unchanged on exit.
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the equations to be solved as
* follows:
*
* TRANS = 'N' or 'n' A*x = b.
*
* TRANS = 'T' or 't' A'*x = b.
*
* TRANS = 'C' or 'c' A'*x = b.
*
* Unchanged on exit.
*
* DIAG - CHARACTER*1.
* On entry, DIAG specifies whether or not A is unit
* triangular as follows:
*
* DIAG = 'U' or 'u' A is assumed to be unit triangular.
*
* DIAG = 'N' or 'n' A is not assumed to be unit
* triangular.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* K - INTEGER.
* On entry with UPLO = 'U' or 'u', K specifies the number of
* super-diagonals of the matrix A.
* On entry with UPLO = 'L' or 'l', K specifies the number of
* sub-diagonals of the matrix A.
* K must satisfy 0 .le. K.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
* Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
* by n part of the array A must contain the upper triangular
* band part of the matrix of coefficients, supplied column by
* column, with the leading diagonal of the matrix in row
* ( k + 1 ) of the array, the first super-diagonal starting at
* position 2 in row k, and so on. The top left k by k triangle
* of the array A is not referenced.
* The following program segment will transfer an upper
* triangular band matrix from conventional full matrix storage
* to band storage:
*
* DO 20, J = 1, N
* M = K + 1 - J
* DO 10, I = MAX( 1, J - K ), J
* A( M + I, J ) = matrix( I, J )
* 10 CONTINUE
* 20 CONTINUE
*
* Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
* by n part of the array A must contain the lower triangular
* band part of the matrix of coefficients, supplied column by
* column, with the leading diagonal of the matrix in row 1 of
* the array, the first sub-diagonal starting at position 1 in
* row 2, and so on. The bottom right k by k triangle of the
* array A is not referenced.
* The following program segment will transfer a lower
* triangular band matrix from conventional full matrix storage
* to band storage:
*
* DO 20, J = 1, N
* M = 1 - J
* DO 10, I = J, MIN( N, J + K )
* A( M + I, J ) = matrix( I, J )
* 10 CONTINUE
* 20 CONTINUE
*
* Note that when DIAG = 'U' or 'u' the elements of the array A
* corresponding to the diagonal elements of the matrix are not
* referenced, but are assumed to be unity.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* ( k + 1 ).
* Unchanged on exit.
*
* X - DOUBLE PRECISION array of dimension at least
* ( 1 + ( n - 1 )*abs( INCX ) ).
* Before entry, the incremented array X must contain the n
* element right-hand side vector b. On exit, X is overwritten
* with the solution vector x.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER (ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L
LOGICAL NOUNIT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX,MIN
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN
INFO = 1
ELSE IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
+ .NOT.LSAME(TRANS,'C')) THEN
INFO = 2
ELSE IF (.NOT.LSAME(DIAG,'U') .AND. .NOT.LSAME(DIAG,'N')) THEN
INFO = 3
ELSE IF (N.LT.0) THEN
INFO = 4
ELSE IF (K.LT.0) THEN
INFO = 5
ELSE IF (LDA.LT. (K+1)) THEN
INFO = 7
ELSE IF (INCX.EQ.0) THEN
INFO = 9
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DTBSV ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF (N.EQ.0) RETURN
*
NOUNIT = LSAME(DIAG,'N')
*
* Set up the start point in X if the increment is not unity. This
* will be ( N - 1 )*INCX too small for descending loops.
*
IF (INCX.LE.0) THEN
KX = 1 - (N-1)*INCX
ELSE IF (INCX.NE.1) THEN
KX = 1
END IF
*
* Start the operations. In this version the elements of A are
* accessed by sequentially with one pass through A.
*
IF (LSAME(TRANS,'N')) THEN
*
* Form x := inv( A )*x.
*
IF (LSAME(UPLO,'U')) THEN
KPLUS1 = K + 1
IF (INCX.EQ.1) THEN
DO 20 J = N,1,-1
IF (X(J).NE.ZERO) THEN
L = KPLUS1 - J
IF (NOUNIT) X(J) = X(J)/A(KPLUS1,J)
TEMP = X(J)
DO 10 I = J - 1,MAX(1,J-K),-1
X(I) = X(I) - TEMP*A(L+I,J)
10 CONTINUE
END IF
20 CONTINUE
ELSE
KX = KX + (N-1)*INCX
JX = KX
DO 40 J = N,1,-1
KX = KX - INCX
IF (X(JX).NE.ZERO) THEN
IX = KX
L = KPLUS1 - J
IF (NOUNIT) X(JX) = X(JX)/A(KPLUS1,J)
TEMP = X(JX)
DO 30 I = J - 1,MAX(1,J-K),-1
X(IX) = X(IX) - TEMP*A(L+I,J)
IX = IX - INCX
30 CONTINUE
END IF
JX = JX - INCX
40 CONTINUE
END IF
ELSE
IF (INCX.EQ.1) THEN
DO 60 J = 1,N
IF (X(J).NE.ZERO) THEN
L = 1 - J
IF (NOUNIT) X(J) = X(J)/A(1,J)
TEMP = X(J)
DO 50 I = J + 1,MIN(N,J+K)
X(I) = X(I) - TEMP*A(L+I,J)
50 CONTINUE
END IF
60 CONTINUE
ELSE
JX = KX
DO 80 J = 1,N
KX = KX + INCX
IF (X(JX).NE.ZERO) THEN
IX = KX
L = 1 - J
IF (NOUNIT) X(JX) = X(JX)/A(1,J)
TEMP = X(JX)
DO 70 I = J + 1,MIN(N,J+K)
X(IX) = X(IX) - TEMP*A(L+I,J)
IX = IX + INCX
70 CONTINUE
END IF
JX = JX + INCX
80 CONTINUE
END IF
END IF
ELSE
*
* Form x := inv( A')*x.
*
IF (LSAME(UPLO,'U')) THEN
KPLUS1 = K + 1
IF (INCX.EQ.1) THEN
DO 100 J = 1,N
TEMP = X(J)
L = KPLUS1 - J
DO 90 I = MAX(1,J-K),J - 1
TEMP = TEMP - A(L+I,J)*X(I)
90 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
X(J) = TEMP
100 CONTINUE
ELSE
JX = KX
DO 120 J = 1,N
TEMP = X(JX)
IX = KX
L = KPLUS1 - J
DO 110 I = MAX(1,J-K),J - 1
TEMP = TEMP - A(L+I,J)*X(IX)
IX = IX + INCX
110 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J)
X(JX) = TEMP
JX = JX + INCX
IF (J.GT.K) KX = KX + INCX
120 CONTINUE
END IF
ELSE
IF (INCX.EQ.1) THEN
DO 140 J = N,1,-1
TEMP = X(J)
L = 1 - J
DO 130 I = MIN(N,J+K),J + 1,-1
TEMP = TEMP - A(L+I,J)*X(I)
130 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(1,J)
X(J) = TEMP
140 CONTINUE
ELSE
KX = KX + (N-1)*INCX
JX = KX
DO 160 J = N,1,-1
TEMP = X(JX)
IX = KX
L = 1 - J
DO 150 I = MIN(N,J+K),J + 1,-1
TEMP = TEMP - A(L+I,J)*X(IX)
IX = IX - INCX
150 CONTINUE
IF (NOUNIT) TEMP = TEMP/A(1,J)
X(JX) = TEMP
JX = JX - INCX
IF ((N-J).GE.K) KX = KX - INCX
160 CONTINUE
END IF
END IF
END IF
*
RETURN
*
* End of DTBSV .
*
END
SUBROUTINE DGEMV(TRANS,M,N,ALPHA,A,LDA,X,INCX,BETA,Y,INCY)
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER INCX,INCY,LDA,M,N
CHARACTER TRANS
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),X(*),Y(*)
* ..
*
* Purpose
* =======
*
* DGEMV performs one of the matrix-vector operations
*
* y := alpha*A*x + beta*y, or y := alpha*A'*x + beta*y,
*
* where alpha and beta are scalars, x and y are vectors and A is an
* m by n matrix.
*
* Arguments
* ==========
*
* TRANS - CHARACTER*1.
* On entry, TRANS specifies the operation to be performed as
* follows:
*
* TRANS = 'N' or 'n' y := alpha*A*x + beta*y.
*
* TRANS = 'T' or 't' y := alpha*A'*x + beta*y.
*
* TRANS = 'C' or 'c' y := alpha*A'*x + beta*y.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix A.
* M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* ALPHA - DOUBLE PRECISION.
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of DIMENSION ( LDA, n ).
* Before entry, the leading m by n part of the array A must
* contain the matrix of coefficients.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* max( 1, m ).
* Unchanged on exit.
*
* X - DOUBLE PRECISION array of DIMENSION at least
* ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = 'N' or 'n'
* and at least
* ( 1 + ( m - 1 )*abs( INCX ) ) otherwise.
* Before entry, the incremented array X must contain the
* vector x.
* Unchanged on exit.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
* BETA - DOUBLE PRECISION.
* On entry, BETA specifies the scalar beta. When BETA is
* supplied as zero then Y need not be set on input.
* Unchanged on exit.
*
* Y - DOUBLE PRECISION array of DIMENSION at least
* ( 1 + ( m - 1 )*abs( INCY ) ) when TRANS = 'N' or 'n'
* and at least
* ( 1 + ( n - 1 )*abs( INCY ) ) otherwise.
* Before entry with BETA non-zero, the incremented array Y
* must contain the vector y. On exit, Y is overwritten by the
* updated vector y.
*
* INCY - INTEGER.
* On entry, INCY specifies the increment for the elements of
* Y. INCY must not be zero.
* Unchanged on exit.
*
*
* Level 2 Blas routine.
*
* -- Written on 22-October-1986.
* Jack Dongarra, Argonne National Lab.
* Jeremy Du Croz, Nag Central Office.
* Sven Hammarling, Nag Central Office.
* Richard Hanson, Sandia National Labs.
*
*
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,IX,IY,J,JX,JY,KX,KY,LENX,LENY
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
*
* Test the input parameters.
*
INFO = 0
IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND.
+ .NOT.LSAME(TRANS,'C')) THEN
INFO = 1
ELSE IF (M.LT.0) THEN
INFO = 2
ELSE IF (N.LT.0) THEN
INFO = 3
ELSE IF (LDA.LT.MAX(1,M)) THEN
INFO = 6
ELSE IF (INCX.EQ.0) THEN
INFO = 8
ELSE IF (INCY.EQ.0) THEN
INFO = 11
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DGEMV ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((M.EQ.0) .OR. (N.EQ.0) .OR.
+ ((ALPHA.EQ.ZERO).AND. (BETA.EQ.ONE))) RETURN
*
* Set LENX and LENY, the lengths of the vectors x and y, and set
* up the start points in X and Y.
*
IF (LSAME(TRANS,'N')) THEN
LENX = N
LENY = M
ELSE
LENX = M
LENY = N
END IF
IF (INCX.GT.0) THEN
KX = 1
ELSE
KX = 1 - (LENX-1)*INCX
END IF
IF (INCY.GT.0) THEN
KY = 1
ELSE
KY = 1 - (LENY-1)*INCY
END IF
*
* Start the operations. In this version the elements of A are
* accessed sequentially with one pass through A.
*
* First form y := beta*y.
*
IF (BETA.NE.ONE) THEN
IF (INCY.EQ.1) THEN
IF (BETA.EQ.ZERO) THEN
DO 10 I = 1,LENY
Y(I) = ZERO
10 CONTINUE
ELSE
DO 20 I = 1,LENY
Y(I) = BETA*Y(I)
20 CONTINUE
END IF
ELSE
IY = KY
IF (BETA.EQ.ZERO) THEN
DO 30 I = 1,LENY
Y(IY) = ZERO
IY = IY + INCY
30 CONTINUE
ELSE
DO 40 I = 1,LENY
Y(IY) = BETA*Y(IY)
IY = IY + INCY
40 CONTINUE
END IF
END IF
END IF
IF (ALPHA.EQ.ZERO) RETURN
IF (LSAME(TRANS,'N')) THEN
*
* Form y := alpha*A*x + y.
*
JX = KX
IF (INCY.EQ.1) THEN
DO 60 J = 1,N
IF (X(JX).NE.ZERO) THEN
TEMP = ALPHA*X(JX)
DO 50 I = 1,M
Y(I) = Y(I) + TEMP*A(I,J)
50 CONTINUE
END IF
JX = JX + INCX
60 CONTINUE
ELSE
DO 80 J = 1,N
IF (X(JX).NE.ZERO) THEN
TEMP = ALPHA*X(JX)
IY = KY
DO 70 I = 1,M
Y(IY) = Y(IY) + TEMP*A(I,J)
IY = IY + INCY
70 CONTINUE
END IF
JX = JX + INCX
80 CONTINUE
END IF
ELSE
*
* Form y := alpha*A'*x + y.
*
JY = KY
IF (INCX.EQ.1) THEN
DO 100 J = 1,N
TEMP = ZERO
DO 90 I = 1,M
TEMP = TEMP + A(I,J)*X(I)
90 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP
JY = JY + INCY
100 CONTINUE
ELSE
DO 120 J = 1,N
TEMP = ZERO
IX = KX
DO 110 I = 1,M
TEMP = TEMP + A(I,J)*X(IX)
IX = IX + INCX
110 CONTINUE
Y(JY) = Y(JY) + ALPHA*TEMP
JY = JY + INCY
120 CONTINUE
END IF
END IF
*
RETURN
*
* End of DGEMV .
*
END
SUBROUTINE DGEMM(TRANSA,TRANSB,M,N,K,ALPHA,A,LDA,B,LDB,BETA,C,LDC)
* .. Scalar Arguments ..
DOUBLE PRECISION ALPHA,BETA
INTEGER K,LDA,LDB,LDC,M,N
CHARACTER TRANSA,TRANSB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A(LDA,*),B(LDB,*),C(LDC,*)
* ..
*
* Purpose
* =======
*
* DGEMM performs one of the matrix-matrix operations
*
* C := alpha*op( A )*op( B ) + beta*C,
*
* where op( X ) is one of
*
* op( X ) = X or op( X ) = X',
*
* alpha and beta are scalars, and A, B and C are matrices, with op( A )
* an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
*
* Arguments
* ==========
*
* TRANSA - CHARACTER*1.
* On entry, TRANSA specifies the form of op( A ) to be used in
* the matrix multiplication as follows:
*
* TRANSA = 'N' or 'n', op( A ) = A.
*
* TRANSA = 'T' or 't', op( A ) = A'.
*
* TRANSA = 'C' or 'c', op( A ) = A'.
*
* Unchanged on exit.
*
* TRANSB - CHARACTER*1.
* On entry, TRANSB specifies the form of op( B ) to be used in
* the matrix multiplication as follows:
*
* TRANSB = 'N' or 'n', op( B ) = B.
*
* TRANSB = 'T' or 't', op( B ) = B'.
*
* TRANSB = 'C' or 'c', op( B ) = B'.
*
* Unchanged on exit.
*
* M - INTEGER.
* On entry, M specifies the number of rows of the matrix
* op( A ) and of the matrix C. M must be at least zero.
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the number of columns of the matrix
* op( B ) and the number of columns of the matrix C. N must be
* at least zero.
* Unchanged on exit.
*
* K - INTEGER.
* On entry, K specifies the number of columns of the matrix
* op( A ) and the number of rows of the matrix op( B ). K must
* be at least zero.
* Unchanged on exit.
*
* ALPHA - DOUBLE PRECISION.
* On entry, ALPHA specifies the scalar alpha.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of DIMENSION ( LDA, ka ), where ka is
* k when TRANSA = 'N' or 'n', and is m otherwise.
* Before entry with TRANSA = 'N' or 'n', the leading m by k
* part of the array A must contain the matrix A, otherwise
* the leading k by m part of the array A must contain the
* matrix A.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. When TRANSA = 'N' or 'n' then
* LDA must be at least max( 1, m ), otherwise LDA must be at
* least max( 1, k ).
* Unchanged on exit.
*
* B - DOUBLE PRECISION array of DIMENSION ( LDB, kb ), where kb is
* n when TRANSB = 'N' or 'n', and is k otherwise.
* Before entry with TRANSB = 'N' or 'n', the leading k by n
* part of the array B must contain the matrix B, otherwise
* the leading n by k part of the array B must contain the
* matrix B.
* Unchanged on exit.
*
* LDB - INTEGER.
* On entry, LDB specifies the first dimension of B as declared
* in the calling (sub) program. When TRANSB = 'N' or 'n' then
* LDB must be at least max( 1, k ), otherwise LDB must be at
* least max( 1, n ).
* Unchanged on exit.
*
* BETA - DOUBLE PRECISION.
* On entry, BETA specifies the scalar beta. When BETA is
* supplied as zero then C need not be set on input.
* Unchanged on exit.
*
* C - DOUBLE PRECISION array of DIMENSION ( LDC, n ).
* Before entry, the leading m by n part of the array C must
* contain the matrix C, except when beta is zero, in which
* case C need not be set on entry.
* On exit, the array C is overwritten by the m by n matrix
* ( alpha*op( A )*op( B ) + beta*C ).
*
* LDC - INTEGER.
* On entry, LDC specifies the first dimension of C as declared
* in the calling (sub) program. LDC must be at least
* max( 1, m ).
* Unchanged on exit.
*
*
* Level 3 Blas routine.
*
* -- Written on 8-February-1989.
* Jack Dongarra, Argonne National Laboratory.
* Iain Duff, AERE Harwell.
* Jeremy Du Croz, Numerical Algorithms Group Ltd.
* Sven Hammarling, Numerical Algorithms Group Ltd.
*
*
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Local Scalars ..
DOUBLE PRECISION TEMP
INTEGER I,INFO,J,L,NCOLA,NROWA,NROWB
LOGICAL NOTA,NOTB
* ..
* .. Parameters ..
DOUBLE PRECISION ONE,ZERO
PARAMETER (ONE=1.0D+0,ZERO=0.0D+0)
* ..
*
* Set NOTA and NOTB as true if A and B respectively are not
* transposed and set NROWA, NCOLA and NROWB as the number of rows
* and columns of A and the number of rows of B respectively.
*
NOTA = LSAME(TRANSA,'N')
NOTB = LSAME(TRANSB,'N')
IF (NOTA) THEN
NROWA = M
NCOLA = K
ELSE
NROWA = K
NCOLA = M
END IF
IF (NOTB) THEN
NROWB = K
ELSE
NROWB = N
END IF
*
* Test the input parameters.
*
INFO = 0
IF ((.NOT.NOTA) .AND. (.NOT.LSAME(TRANSA,'C')) .AND.
+ (.NOT.LSAME(TRANSA,'T'))) THEN
INFO = 1
ELSE IF ((.NOT.NOTB) .AND. (.NOT.LSAME(TRANSB,'C')) .AND.
+ (.NOT.LSAME(TRANSB,'T'))) THEN
INFO = 2
ELSE IF (M.LT.0) THEN
INFO = 3
ELSE IF (N.LT.0) THEN
INFO = 4
ELSE IF (K.LT.0) THEN
INFO = 5
ELSE IF (LDA.LT.MAX(1,NROWA)) THEN
INFO = 8
ELSE IF (LDB.LT.MAX(1,NROWB)) THEN
INFO = 10
ELSE IF (LDC.LT.MAX(1,M)) THEN
INFO = 13
END IF
IF (INFO.NE.0) THEN
CALL XERBLA('DGEMM ',INFO)
RETURN
END IF
*
* Quick return if possible.
*
IF ((M.EQ.0) .OR. (N.EQ.0) .OR.
+ (((ALPHA.EQ.ZERO).OR. (K.EQ.0)).AND. (BETA.EQ.ONE))) RETURN
*
* And if alpha.eq.zero.
*
IF (ALPHA.EQ.ZERO) THEN
IF (BETA.EQ.ZERO) THEN
DO 20 J = 1,N
DO 10 I = 1,M
C(I,J) = ZERO
10 CONTINUE
20 CONTINUE
ELSE
DO 40 J = 1,N
DO 30 I = 1,M
C(I,J) = BETA*C(I,J)
30 CONTINUE
40 CONTINUE
END IF
RETURN
END IF
*
* Start the operations.
*
IF (NOTB) THEN
IF (NOTA) THEN
*
* Form C := alpha*A*B + beta*C.
*
DO 90 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 50 I = 1,M
C(I,J) = ZERO
50 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 60 I = 1,M
C(I,J) = BETA*C(I,J)
60 CONTINUE
END IF
DO 80 L = 1,K
IF (B(L,J).NE.ZERO) THEN
TEMP = ALPHA*B(L,J)
DO 70 I = 1,M
C(I,J) = C(I,J) + TEMP*A(I,L)
70 CONTINUE
END IF
80 CONTINUE
90 CONTINUE
ELSE
*
* Form C := alpha*A'*B + beta*C
*
DO 120 J = 1,N
DO 110 I = 1,M
TEMP = ZERO
DO 100 L = 1,K
TEMP = TEMP + A(L,I)*B(L,J)
100 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP
ELSE
C(I,J) = ALPHA*TEMP + BETA*C(I,J)
END IF
110 CONTINUE
120 CONTINUE
END IF
ELSE
IF (NOTA) THEN
*
* Form C := alpha*A*B' + beta*C
*
DO 170 J = 1,N
IF (BETA.EQ.ZERO) THEN
DO 130 I = 1,M
C(I,J) = ZERO
130 CONTINUE
ELSE IF (BETA.NE.ONE) THEN
DO 140 I = 1,M
C(I,J) = BETA*C(I,J)
140 CONTINUE
END IF
DO 160 L = 1,K
IF (B(J,L).NE.ZERO) THEN
TEMP = ALPHA*B(J,L)
DO 150 I = 1,M
C(I,J) = C(I,J) + TEMP*A(I,L)
150 CONTINUE
END IF
160 CONTINUE
170 CONTINUE
ELSE
*
* Form C := alpha*A'*B' + beta*C
*
DO 200 J = 1,N
DO 190 I = 1,M
TEMP = ZERO
DO 180 L = 1,K
TEMP = TEMP + A(L,I)*B(J,L)
180 CONTINUE
IF (BETA.EQ.ZERO) THEN
C(I,J) = ALPHA*TEMP
ELSE
C(I,J) = ALPHA*TEMP + BETA*C(I,J)
END IF
190 CONTINUE
200 CONTINUE
END IF
END IF
*
RETURN
*
* End of DGEMM .
*
END
INTEGER FUNCTION IDAMAX(N,DX,INCX)
* .. Scalar Arguments ..
INTEGER INCX,N
* ..
* .. Array Arguments ..
DOUBLE PRECISION DX(*)
* ..
*
* Purpose
* =======
*
* finds the index of element having max. absolute value.
* jack dongarra, linpack, 3/11/78.
* modified 3/93 to return if incx .le. 0.
* modified 12/3/93, array(1) declarations changed to array(*)
*
*
* .. Local Scalars ..
DOUBLE PRECISION DMAX
INTEGER I,IX
* ..
* .. Intrinsic Functions ..
INTRINSIC DABS
* ..
IDAMAX = 0
IF (N.LT.1 .OR. INCX.LE.0) RETURN
IDAMAX = 1
IF (N.EQ.1) RETURN
IF (INCX.EQ.1) GO TO 20
*
* code for increment not equal to 1
*
IX = 1
DMAX = DABS(DX(1))
IX = IX + INCX
DO 10 I = 2,N
IF (DABS(DX(IX)).LE.DMAX) GO TO 5
IDAMAX = I
DMAX = DABS(DX(IX))
5 IX = IX + INCX
10 CONTINUE
RETURN
*
* code for increment equal to 1
*
20 DMAX = DABS(DX(1))
DO 30 I = 2,N
IF (DABS(DX(I)).LE.DMAX) GO TO 30
IDAMAX = I
DMAX = DABS(DX(I))
30 CONTINUE
RETURN
END
LOGICAL FUNCTION LSAME(CA,CB)
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER CA,CB
* ..
*
* Purpose
* =======
*
* LSAME returns .TRUE. if CA is the same letter as CB regardless of
* case.
*
* Arguments
* =========
*
* CA (input) CHARACTER*1
*
* CB (input) CHARACTER*1
* CA and CB specify the single characters to be compared.
*
* =====================================================================
*
* .. Intrinsic Functions ..
INTRINSIC ICHAR
* ..
* .. Local Scalars ..
INTEGER INTA,INTB,ZCODE
* ..
*
* Test if the characters are equal
*
LSAME = CA .EQ. CB
IF (LSAME) RETURN
*
* Now test for equivalence if both characters are alphabetic.
*
ZCODE = ICHAR('Z')
*
* Use 'Z' rather than 'A' so that ASCII can be detected on Prime
* machines, on which ICHAR returns a value with bit 8 set.
* ICHAR('A') on Prime machines returns 193 which is the same as
* ICHAR('A') on an EBCDIC machine.
*
INTA = ICHAR(CA)
INTB = ICHAR(CB)
*
IF (ZCODE.EQ.90 .OR. ZCODE.EQ.122) THEN
*
* ASCII is assumed - ZCODE is the ASCII code of either lower or
* upper case 'Z'.
*
IF (INTA.GE.97 .AND. INTA.LE.122) INTA = INTA - 32
IF (INTB.GE.97 .AND. INTB.LE.122) INTB = INTB - 32
*
ELSE IF (ZCODE.EQ.233 .OR. ZCODE.EQ.169) THEN
*
* EBCDIC is assumed - ZCODE is the EBCDIC code of either lower or
* upper case 'Z'.
*
IF (INTA.GE.129 .AND. INTA.LE.137 .OR.
+ INTA.GE.145 .AND. INTA.LE.153 .OR.
+ INTA.GE.162 .AND. INTA.LE.169) INTA = INTA + 64
IF (INTB.GE.129 .AND. INTB.LE.137 .OR.
+ INTB.GE.145 .AND. INTB.LE.153 .OR.
+ INTB.GE.162 .AND. INTB.LE.169) INTB = INTB + 64
*
ELSE IF (ZCODE.EQ.218 .OR. ZCODE.EQ.250) THEN
*
* ASCII is assumed, on Prime machines - ZCODE is the ASCII code
* plus 128 of either lower or upper case 'Z'.
*
IF (INTA.GE.225 .AND. INTA.LE.250) INTA = INTA - 32
IF (INTB.GE.225 .AND. INTB.LE.250) INTB = INTB - 32
END IF
LSAME = INTA .EQ. INTB
*
* RETURN
*
* End of LSAME
*
END
DOUBLE PRECISION FUNCTION DLAMCH( CMACH )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
CHARACTER CMACH
* ..
*
* Purpose
* =======
*
* DLAMCH determines double precision machine parameters.
*
* Arguments
* =========
*
* CMACH (input) CHARACTER*1
* Specifies the value to be returned by DLAMCH:
* = 'E' or 'e', DLAMCH := eps
* = 'S' or 's , DLAMCH := sfmin
* = 'B' or 'b', DLAMCH := base
* = 'P' or 'p', DLAMCH := eps*base
* = 'N' or 'n', DLAMCH := t
* = 'R' or 'r', DLAMCH := rnd
* = 'M' or 'm', DLAMCH := emin
* = 'U' or 'u', DLAMCH := rmin
* = 'L' or 'l', DLAMCH := emax
* = 'O' or 'o', DLAMCH := rmax
*
* where
*
* eps = relative machine precision
* sfmin = safe minimum, such that 1/sfmin does not overflow
* base = base of the machine
* prec = eps*base
* t = number of (base) digits in the mantissa
* rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
* emin = minimum exponent before (gradual) underflow
* rmin = underflow threshold - base**(emin-1)
* emax = largest exponent before overflow
* rmax = overflow threshold - (base**emax)*(1-eps)
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL FIRST, LRND
INTEGER BETA, IMAX, IMIN, IT
DOUBLE PRECISION BASE, EMAX, EMIN, EPS, PREC, RMACH, RMAX, RMIN,
$ RND, SFMIN, SMALL, T
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL DLAMC2
* ..
* .. Save statement ..
SAVE FIRST, EPS, SFMIN, BASE, T, RND, EMIN, RMIN,
$ EMAX, RMAX, PREC
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
CALL DLAMC2( BETA, IT, LRND, EPS, IMIN, RMIN, IMAX, RMAX )
BASE = BETA
T = IT
IF( LRND ) THEN
RND = ONE
EPS = ( BASE**( 1-IT ) ) / 2
ELSE
RND = ZERO
EPS = BASE**( 1-IT )
END IF
PREC = EPS*BASE
EMIN = IMIN
EMAX = IMAX
SFMIN = RMIN
SMALL = ONE / RMAX
IF( SMALL.GE.SFMIN ) THEN
*
* Use SMALL plus a bit, to avoid the possibility of rounding
* causing overflow when computing 1/sfmin.
*
SFMIN = SMALL*( ONE+EPS )
END IF
END IF
*
IF( LSAME( CMACH, 'E' ) ) THEN
RMACH = EPS
ELSE IF( LSAME( CMACH, 'S' ) ) THEN
RMACH = SFMIN
ELSE IF( LSAME( CMACH, 'B' ) ) THEN
RMACH = BASE
ELSE IF( LSAME( CMACH, 'P' ) ) THEN
RMACH = PREC
ELSE IF( LSAME( CMACH, 'N' ) ) THEN
RMACH = T
ELSE IF( LSAME( CMACH, 'R' ) ) THEN
RMACH = RND
ELSE IF( LSAME( CMACH, 'M' ) ) THEN
RMACH = EMIN
ELSE IF( LSAME( CMACH, 'U' ) ) THEN
RMACH = RMIN
ELSE IF( LSAME( CMACH, 'L' ) ) THEN
RMACH = EMAX
ELSE IF( LSAME( CMACH, 'O' ) ) THEN
RMACH = RMAX
END IF
*
DLAMCH = RMACH
RETURN
*
* End of DLAMCH
*
END
*
************************************************************************
*
SUBROUTINE DLAMC1( BETA, T, RND, IEEE1 )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL IEEE1, RND
INTEGER BETA, T
* ..
*
* Purpose
* =======
*
* DLAMC1 determines the machine parameters given by BETA, T, RND, and
* IEEE1.
*
* Arguments
* =========
*
* BETA (output) INTEGER
* The base of the machine.
*
* T (output) INTEGER
* The number of ( BETA ) digits in the mantissa.
*
* RND (output) LOGICAL
* Specifies whether proper rounding ( RND = .TRUE. ) or
* chopping ( RND = .FALSE. ) occurs in addition. This may not
* be a reliable guide to the way in which the machine performs
* its arithmetic.
*
* IEEE1 (output) LOGICAL
* Specifies whether rounding appears to be done in the IEEE
* 'round to nearest' style.
*
* Further Details
* ===============
*
* The routine is based on the routine ENVRON by Malcolm and
* incorporates suggestions by Gentleman and Marovich. See
*
* Malcolm M. A. (1972) Algorithms to reveal properties of
* floating-point arithmetic. Comms. of the ACM, 15, 949-951.
*
* Gentleman W. M. and Marovich S. B. (1974) More on algorithms
* that reveal properties of floating point arithmetic units.
* Comms. of the ACM, 17, 276-277.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL FIRST, LIEEE1, LRND
INTEGER LBETA, LT
DOUBLE PRECISION A, B, C, F, ONE, QTR, SAVEC, T1, T2
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Save statement ..
SAVE FIRST, LIEEE1, LBETA, LRND, LT
* ..
* .. Data statements ..
DATA FIRST / .TRUE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ONE = 1
*
* LBETA, LIEEE1, LT and LRND are the local values of BETA,
* IEEE1, T and RND.
*
* Throughout this routine we use the function DLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* Compute a = 2.0**m with the smallest positive integer m such
* that
*
* fl( a + 1.0 ) = a.
*
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
10 CONTINUE
IF( C.EQ.ONE ) THEN
A = 2*A
C = DLAMC3( A, ONE )
C = DLAMC3( C, -A )
GO TO 10
END IF
*+ END WHILE
*
* Now compute b = 2.0**m with the smallest positive integer m
* such that
*
* fl( a + b ) .gt. a.
*
B = 1
C = DLAMC3( A, B )
*
*+ WHILE( C.EQ.A )LOOP
20 CONTINUE
IF( C.EQ.A ) THEN
B = 2*B
C = DLAMC3( A, B )
GO TO 20
END IF
*+ END WHILE
*
* Now compute the base. a and c are neighbouring floating point
* numbers in the interval ( beta**t, beta**( t + 1 ) ) and so
* their difference is beta. Adding 0.25 to c is to ensure that it
* is truncated to beta and not ( beta - 1 ).
*
QTR = ONE / 4
SAVEC = C
C = DLAMC3( C, -A )
LBETA = C + QTR
*
* Now determine whether rounding or chopping occurs, by adding a
* bit less than beta/2 and a bit more than beta/2 to a.
*
B = LBETA
F = DLAMC3( B / 2, -B / 100 )
C = DLAMC3( F, A )
IF( C.EQ.A ) THEN
LRND = .TRUE.
ELSE
LRND = .FALSE.
END IF
F = DLAMC3( B / 2, B / 100 )
C = DLAMC3( F, A )
IF( ( LRND ) .AND. ( C.EQ.A ) )
$ LRND = .FALSE.
*
* Try and decide whether rounding is done in the IEEE 'round to
* nearest' style. B/2 is half a unit in the last place of the two
* numbers A and SAVEC. Furthermore, A is even, i.e. has last bit
* zero, and SAVEC is odd. Thus adding B/2 to A should not change
* A, but adding B/2 to SAVEC should change SAVEC.
*
T1 = DLAMC3( B / 2, A )
T2 = DLAMC3( B / 2, SAVEC )
LIEEE1 = ( T1.EQ.A ) .AND. ( T2.GT.SAVEC ) .AND. LRND
*
* Now find the mantissa, t. It should be the integer part of
* log to the base beta of a, however it is safer to determine t
* by powering. So we find t as the smallest positive integer for
* which
*
* fl( beta**t + 1.0 ) = 1.0.
*
LT = 0
A = 1
C = 1
*
*+ WHILE( C.EQ.ONE )LOOP
30 CONTINUE
IF( C.EQ.ONE ) THEN
LT = LT + 1
A = A*LBETA
C = DLAMC3( A, ONE )
C = DLAMC3( C, -A )
GO TO 30
END IF
*+ END WHILE
*
END IF
*
BETA = LBETA
T = LT
RND = LRND
IEEE1 = LIEEE1
RETURN
*
* End of DLAMC1
*
END
*
************************************************************************
*
SUBROUTINE DLAMC2( BETA, T, RND, EPS, EMIN, RMIN, EMAX, RMAX )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL RND
INTEGER BETA, EMAX, EMIN, T
DOUBLE PRECISION EPS, RMAX, RMIN
* ..
*
* Purpose
* =======
*
* DLAMC2 determines the machine parameters specified in its argument
* list.
*
* Arguments
* =========
*
* BETA (output) INTEGER
* The base of the machine.
*
* T (output) INTEGER
* The number of ( BETA ) digits in the mantissa.
*
* RND (output) LOGICAL
* Specifies whether proper rounding ( RND = .TRUE. ) or
* chopping ( RND = .FALSE. ) occurs in addition. This may not
* be a reliable guide to the way in which the machine performs
* its arithmetic.
*
* EPS (output) DOUBLE PRECISION
* The smallest positive number such that
*
* fl( 1.0 - EPS ) .LT. 1.0,
*
* where fl denotes the computed value.
*
* EMIN (output) INTEGER
* The minimum exponent before (gradual) underflow occurs.
*
* RMIN (output) DOUBLE PRECISION
* The smallest normalized number for the machine, given by
* BASE**( EMIN - 1 ), where BASE is the floating point value
* of BETA.
*
* EMAX (output) INTEGER
* The maximum exponent before overflow occurs.
*
* RMAX (output) DOUBLE PRECISION
* The largest positive number for the machine, given by
* BASE**EMAX * ( 1 - EPS ), where BASE is the floating point
* value of BETA.
*
* Further Details
* ===============
*
* The computation of EPS is based on a routine PARANOIA by
* W. Kahan of the University of California at Berkeley.
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL FIRST, IEEE, IWARN, LIEEE1, LRND
INTEGER GNMIN, GPMIN, I, LBETA, LEMAX, LEMIN, LT,
$ NGNMIN, NGPMIN
DOUBLE PRECISION A, B, C, HALF, LEPS, LRMAX, LRMIN, ONE, RBASE,
$ SIXTH, SMALL, THIRD, TWO, ZERO
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. External Subroutines ..
EXTERNAL DLAMC1, DLAMC4, DLAMC5
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Save statement ..
SAVE FIRST, IWARN, LBETA, LEMAX, LEMIN, LEPS, LRMAX,
$ LRMIN, LT
* ..
* .. Data statements ..
DATA FIRST / .TRUE. / , IWARN / .FALSE. /
* ..
* .. Executable Statements ..
*
IF( FIRST ) THEN
FIRST = .FALSE.
ZERO = 0
ONE = 1
TWO = 2
*
* LBETA, LT, LRND, LEPS, LEMIN and LRMIN are the local values of
* BETA, T, RND, EPS, EMIN and RMIN.
*
* Throughout this routine we use the function DLAMC3 to ensure
* that relevant values are stored and not held in registers, or
* are not affected by optimizers.
*
* DLAMC1 returns the parameters LBETA, LT, LRND and LIEEE1.
*
CALL DLAMC1( LBETA, LT, LRND, LIEEE1 )
*
* Start to find EPS.
*
B = LBETA
A = B**( -LT )
LEPS = A
*
* Try some tricks to see whether or not this is the correct EPS.
*
B = TWO / 3
HALF = ONE / 2
SIXTH = DLAMC3( B, -HALF )
THIRD = DLAMC3( SIXTH, SIXTH )
B = DLAMC3( THIRD, -HALF )
B = DLAMC3( B, SIXTH )
B = ABS( B )
IF( B.LT.LEPS )
$ B = LEPS
*
LEPS = 1
*
*+ WHILE( ( LEPS.GT.B ).AND.( B.GT.ZERO ) )LOOP
10 CONTINUE
IF( ( LEPS.GT.B ) .AND. ( B.GT.ZERO ) ) THEN
LEPS = B
C = DLAMC3( HALF*LEPS, ( TWO**5 )*( LEPS**2 ) )
C = DLAMC3( HALF, -C )
B = DLAMC3( HALF, C )
C = DLAMC3( HALF, -B )
B = DLAMC3( HALF, C )
GO TO 10
END IF
*+ END WHILE
*
IF( A.LT.LEPS )
$ LEPS = A
*
* Computation of EPS complete.
*
* Now find EMIN. Let A = + or - 1, and + or - (1 + BASE**(-3)).
* Keep dividing A by BETA until (gradual) underflow occurs. This
* is detected when we cannot recover the previous A.
*
RBASE = ONE / LBETA
SMALL = ONE
DO 20 I = 1, 3
SMALL = DLAMC3( SMALL*RBASE, ZERO )
20 CONTINUE
A = DLAMC3( ONE, SMALL )
CALL DLAMC4( NGPMIN, ONE, LBETA )
CALL DLAMC4( NGNMIN, -ONE, LBETA )
CALL DLAMC4( GPMIN, A, LBETA )
CALL DLAMC4( GNMIN, -A, LBETA )
IEEE = .FALSE.
*
IF( ( NGPMIN.EQ.NGNMIN ) .AND. ( GPMIN.EQ.GNMIN ) ) THEN
IF( NGPMIN.EQ.GPMIN ) THEN
LEMIN = NGPMIN
* ( Non twos-complement machines, no gradual underflow;
* e.g., VAX )
ELSE IF( ( GPMIN-NGPMIN ).EQ.3 ) THEN
LEMIN = NGPMIN - 1 + LT
IEEE = .TRUE.
* ( Non twos-complement machines, with gradual underflow;
* e.g., IEEE standard followers )
ELSE
LEMIN = MIN( NGPMIN, GPMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( NGPMIN.EQ.GPMIN ) .AND. ( NGNMIN.EQ.GNMIN ) ) THEN
IF( ABS( NGPMIN-NGNMIN ).EQ.1 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN )
* ( Twos-complement machines, no gradual underflow;
* e.g., CYBER 205 )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE IF( ( ABS( NGPMIN-NGNMIN ).EQ.1 ) .AND.
$ ( GPMIN.EQ.GNMIN ) ) THEN
IF( ( GPMIN-MIN( NGPMIN, NGNMIN ) ).EQ.3 ) THEN
LEMIN = MAX( NGPMIN, NGNMIN ) - 1 + LT
* ( Twos-complement machines with gradual underflow;
* no known machine )
ELSE
LEMIN = MIN( NGPMIN, NGNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
*
ELSE
LEMIN = MIN( NGPMIN, NGNMIN, GPMIN, GNMIN )
* ( A guess; no known machine )
IWARN = .TRUE.
END IF
***
* Comment out this if block if EMIN is ok
IF( IWARN ) THEN
FIRST = .TRUE.
WRITE( 6, FMT = 9999 )LEMIN
END IF
***
*
* Assume IEEE arithmetic if we found denormalised numbers above,
* or if arithmetic seems to round in the IEEE style, determined
* in routine DLAMC1. A true IEEE machine should have both things
* true; however, faulty machines may have one or the other.
*
IEEE = IEEE .OR. LIEEE1
*
* Compute RMIN by successive division by BETA. We could compute
* RMIN as BASE**( EMIN - 1 ), but some machines underflow during
* this computation.
*
LRMIN = 1
DO 30 I = 1, 1 - LEMIN
LRMIN = DLAMC3( LRMIN*RBASE, ZERO )
30 CONTINUE
*
* Finally, call DLAMC5 to compute EMAX and RMAX.
*
CALL DLAMC5( LBETA, LT, LEMIN, IEEE, LEMAX, LRMAX )
END IF
*
BETA = LBETA
T = LT
RND = LRND
EPS = LEPS
EMIN = LEMIN
RMIN = LRMIN
EMAX = LEMAX
RMAX = LRMAX
*
RETURN
*
9999 FORMAT( / / ' WARNING. The value EMIN may be incorrect:-',
$ ' EMIN = ', I8, /
$ ' If, after inspection, the value EMIN looks',
$ ' acceptable please comment out ',
$ / ' the IF block as marked within the code of routine',
$ ' DLAMC2,', / ' otherwise supply EMIN explicitly.', / )
*
* End of DLAMC2
*
END
*
************************************************************************
*
DOUBLE PRECISION FUNCTION DLAMC3( A, B )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B
* ..
*
* Purpose
* =======
*
* DLAMC3 is intended to force A and B to be stored prior to doing
* the addition of A and B , for use in situations where optimizers
* might hold one of these in a register.
*
* Arguments
* =========
*
* A, B (input) DOUBLE PRECISION
* The values A and B.
*
* =====================================================================
*
* .. Executable Statements ..
*
DLAMC3 = A + B
*
RETURN
*
* End of DLAMC3
*
END
*
************************************************************************
*
SUBROUTINE DLAMC4( EMIN, START, BASE )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
INTEGER BASE, EMIN
DOUBLE PRECISION START
* ..
*
* Purpose
* =======
*
* DLAMC4 is a service routine for DLAMC2.
*
* Arguments
* =========
*
* EMIN (output) EMIN
* The minimum exponent before (gradual) underflow, computed by
* setting A = START and dividing by BASE until the previous A
* can not be recovered.
*
* START (input) DOUBLE PRECISION
* The starting point for determining EMIN.
*
* BASE (input) INTEGER
* The base of the machine.
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION A, B1, B2, C1, C2, D1, D2, ONE, RBASE, ZERO
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Executable Statements ..
*
A = START
ONE = 1
RBASE = ONE / BASE
ZERO = 0
EMIN = 1
B1 = DLAMC3( A*RBASE, ZERO )
C1 = A
C2 = A
D1 = A
D2 = A
*+ WHILE( ( C1.EQ.A ).AND.( C2.EQ.A ).AND.
* $ ( D1.EQ.A ).AND.( D2.EQ.A ) )LOOP
10 CONTINUE
IF( ( C1.EQ.A ) .AND. ( C2.EQ.A ) .AND. ( D1.EQ.A ) .AND.
$ ( D2.EQ.A ) ) THEN
EMIN = EMIN - 1
A = B1
B1 = DLAMC3( A / BASE, ZERO )
C1 = DLAMC3( B1*BASE, ZERO )
D1 = ZERO
DO 20 I = 1, BASE
D1 = D1 + B1
20 CONTINUE
B2 = DLAMC3( A*RBASE, ZERO )
C2 = DLAMC3( B2 / RBASE, ZERO )
D2 = ZERO
DO 30 I = 1, BASE
D2 = D2 + B2
30 CONTINUE
GO TO 10
END IF
*+ END WHILE
*
RETURN
*
* End of DLAMC4
*
END
*
************************************************************************
*
SUBROUTINE DLAMC5( BETA, P, EMIN, IEEE, EMAX, RMAX )
*
* -- LAPACK auxiliary routine (version 1.1) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* October 31, 1992
*
* .. Scalar Arguments ..
LOGICAL IEEE
INTEGER BETA, EMAX, EMIN, P
DOUBLE PRECISION RMAX
* ..
*
* Purpose
* =======
*
* DLAMC5 attempts to compute RMAX, the largest machine floating-point
* number, without overflow. It assumes that EMAX + abs(EMIN) sum
* approximately to a power of 2. It will fail on machines where this
* assumption does not hold, for example, the Cyber 205 (EMIN = -28625,
* EMAX = 28718). It will also fail if the value supplied for EMIN is
* too large (i.e. too close to zero), probably with overflow.
*
* Arguments
* =========
*
* BETA (input) INTEGER
* The base of floating-point arithmetic.
*
* P (input) INTEGER
* The number of base BETA digits in the mantissa of a
* floating-point value.
*
* EMIN (input) INTEGER
* The minimum exponent before (gradual) underflow.
*
* IEEE (input) LOGICAL
* A logical flag specifying whether or not the arithmetic
* system is thought to comply with the IEEE standard.
*
* EMAX (output) INTEGER
* The largest exponent before overflow
*
* RMAX (output) DOUBLE PRECISION
* The largest machine floating-point number.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER EXBITS, EXPSUM, I, LEXP, NBITS, TRY, UEXP
DOUBLE PRECISION OLDY, RECBAS, Y, Z
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMC3
EXTERNAL DLAMC3
* ..
* .. Intrinsic Functions ..
INTRINSIC MOD
* ..
* .. Executable Statements ..
*
* First compute LEXP and UEXP, two powers of 2 that bound
* abs(EMIN). We then assume that EMAX + abs(EMIN) will sum
* approximately to the bound that is closest to abs(EMIN).
* (EMAX is the exponent of the required number RMAX).
*
LEXP = 1
EXBITS = 1
10 CONTINUE
TRY = LEXP*2
IF( TRY.LE.( -EMIN ) ) THEN
LEXP = TRY
EXBITS = EXBITS + 1
GO TO 10
END IF
IF( LEXP.EQ.-EMIN ) THEN
UEXP = LEXP
ELSE
UEXP = TRY
EXBITS = EXBITS + 1
END IF
*
* Now -LEXP is less than or equal to EMIN, and -UEXP is greater
* than or equal to EMIN. EXBITS is the number of bits needed to
* store the exponent.
*
IF( ( UEXP+EMIN ).GT.( -LEXP-EMIN ) ) THEN
EXPSUM = 2*LEXP
ELSE
EXPSUM = 2*UEXP
END IF
*
* EXPSUM is the exponent range, approximately equal to
* EMAX - EMIN + 1 .
*
EMAX = EXPSUM + EMIN - 1
NBITS = 1 + EXBITS + P
*
* NBITS is the total number of bits needed to store a
* floating-point number.
*
IF( ( MOD( NBITS, 2 ).EQ.1 ) .AND. ( BETA.EQ.2 ) ) THEN
*
* Either there are an odd number of bits used to store a
* floating-point number, which is unlikely, or some bits are
* not used in the representation of numbers, which is possible,
* (e.g. Cray machines) or the mantissa has an implicit bit,
* (e.g. IEEE machines, Dec Vax machines), which is perhaps the
* most likely. We have to assume the last alternative.
* If this is true, then we need to reduce EMAX by one because
* there must be some way of representing zero in an implicit-bit
* system. On machines like Cray, we are reducing EMAX by one
* unnecessarily.
*
EMAX = EMAX - 1
END IF
*
IF( IEEE ) THEN
*
* Assume we are on an IEEE machine which reserves one exponent
* for infinity and NaN.
*
EMAX = EMAX - 1
END IF
*
* Now create RMAX, the largest machine number, which should
* be equal to (1.0 - BETA**(-P)) * BETA**EMAX .
*
* First compute 1.0 - BETA**(-P), being careful that the
* result is less than 1.0 .
*
RECBAS = ONE / BETA
Z = BETA - ONE
Y = ZERO
DO 20 I = 1, P
Z = Z*RECBAS
IF( Y.LT.ONE )
$ OLDY = Y
Y = DLAMC3( Y, Z )
20 CONTINUE
IF( Y.GE.ONE )
$ Y = OLDY
*
* Now multiply by BETA**EMAX to get RMAX.
*
DO 30 I = 1, EMAX
Y = DLAMC3( Y*BETA, ZERO )
30 CONTINUE
*
RMAX = Y
RETURN
*
* End of DLAMC5
*
END
INTEGER FUNCTION ILAENV( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER*( * ) NAME, OPTS
INTEGER ISPEC, N1, N2, N3, N4
* ..
*
* Purpose
* =======
*
* ILAENV is called from the LAPACK routines to choose problem-dependent
* parameters for the local environment. See ISPEC for a description of
* the parameters.
*
* This version provides a set of parameters which should give good,
* but not optimal, performance on many of the currently available
* computers. Users are encouraged to modify this subroutine to set
* the tuning parameters for their particular machine using the option
* and problem size information in the arguments.
*
* This routine will not function correctly if it is converted to all
* lower case. Converting it to all upper case is allowed.
*
* Arguments
* =========
*
* ISPEC (input) INTEGER
* Specifies the parameter to be returned as the value of
* ILAENV.
* = 1: the optimal blocksize; if this value is 1, an unblocked
* algorithm will give the best performance.
* = 2: the minimum block size for which the block routine
* should be used; if the usable block size is less than
* this value, an unblocked routine should be used.
* = 3: the crossover point (in a block routine, for N less
* than this value, an unblocked routine should be used)
* = 4: the number of shifts, used in the nonsymmetric
* eigenvalue routines (DEPRECATED)
* = 5: the minimum column dimension for blocking to be used;
* rectangular blocks must have dimension at least k by m,
* where k is given by ILAENV(2,...) and m by ILAENV(5,...)
* = 6: the crossover point for the SVD (when reducing an m by n
* matrix to bidiagonal form, if max(m,n)/min(m,n) exceeds
* this value, a QR factorization is used first to reduce
* the matrix to a triangular form.)
* = 7: the number of processors
* = 8: the crossover point for the multishift QR method
* for nonsymmetric eigenvalue problems (DEPRECATED)
* = 9: maximum size of the subproblems at the bottom of the
* computation tree in the divide-and-conquer algorithm
* (used by xGELSD and xGESDD)
* =10: ieee NaN arithmetic can be trusted not to trap
* =11: infinity arithmetic can be trusted not to trap
* 12 <= ISPEC <= 16:
* xHSEQR or one of its subroutines,
* see IPARMQ for detailed explanation
*
* NAME (input) CHARACTER*(*)
* The name of the calling subroutine, in either upper case or
* lower case.
*
* OPTS (input) CHARACTER*(*)
* The character options to the subroutine NAME, concatenated
* into a single character string. For example, UPLO = 'U',
* TRANS = 'T', and DIAG = 'N' for a triangular routine would
* be specified as OPTS = 'UTN'.
*
* N1 (input) INTEGER
* N2 (input) INTEGER
* N3 (input) INTEGER
* N4 (input) INTEGER
* Problem dimensions for the subroutine NAME; these may not all
* be required.
*
* (ILAENV) (output) INTEGER
* >= 0: the value of the parameter specified by ISPEC
* < 0: if ILAENV = -k, the k-th argument had an illegal value.
*
* Further Details
* ===============
*
* The following conventions have been used when calling ILAENV from the
* LAPACK routines:
* 1) OPTS is a concatenation of all of the character options to
* subroutine NAME, in the same order that they appear in the
* argument list for NAME, even if they are not used in determining
* the value of the parameter specified by ISPEC.
* 2) The problem dimensions N1, N2, N3, N4 are specified in the order
* that they appear in the argument list for NAME. N1 is used
* first, N2 second, and so on, and unused problem dimensions are
* passed a value of -1.
* 3) The parameter value returned by ILAENV is checked for validity in
* the calling subroutine. For example, ILAENV is used to retrieve
* the optimal blocksize for STRTRI as follows:
*
* NB = ILAENV( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 )
* IF( NB.LE.1 ) NB = MAX( 1, N )
*
* =====================================================================
*
* .. Local Scalars ..
INTEGER I, IC, IZ, NB, NBMIN, NX
LOGICAL CNAME, SNAME
CHARACTER C1*1, C2*2, C4*2, C3*3, SUBNAM*6
* ..
* .. Intrinsic Functions ..
INTRINSIC CHAR, ICHAR, INT, MIN, REAL
* ..
* .. External Functions ..
INTEGER IEEECK, IPARMQ
EXTERNAL IEEECK, IPARMQ
* ..
* .. Executable Statements ..
*
GO TO ( 10, 10, 10, 80, 90, 100, 110, 120,
$ 130, 140, 150, 160, 160, 160, 160, 160 )ISPEC
*
* Invalid value for ISPEC
*
ILAENV = -1
RETURN
*
10 CONTINUE
*
* Convert NAME to upper case if the first character is lower case.
*
ILAENV = 1
SUBNAM = NAME
IC = ICHAR( SUBNAM( 1: 1 ) )
IZ = ICHAR( 'Z' )
IF( IZ.EQ.90 .OR. IZ.EQ.122 ) THEN
*
* ASCII character set
*
IF( IC.GE.97 .AND. IC.LE.122 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 20 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.97 .AND. IC.LE.122 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
20 CONTINUE
END IF
*
ELSE IF( IZ.EQ.233 .OR. IZ.EQ.169 ) THEN
*
* EBCDIC character set
*
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) ) THEN
SUBNAM( 1: 1 ) = CHAR( IC+64 )
DO 30 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( ( IC.GE.129 .AND. IC.LE.137 ) .OR.
$ ( IC.GE.145 .AND. IC.LE.153 ) .OR.
$ ( IC.GE.162 .AND. IC.LE.169 ) )SUBNAM( I:
$ I ) = CHAR( IC+64 )
30 CONTINUE
END IF
*
ELSE IF( IZ.EQ.218 .OR. IZ.EQ.250 ) THEN
*
* Prime machines: ASCII+128
*
IF( IC.GE.225 .AND. IC.LE.250 ) THEN
SUBNAM( 1: 1 ) = CHAR( IC-32 )
DO 40 I = 2, 6
IC = ICHAR( SUBNAM( I: I ) )
IF( IC.GE.225 .AND. IC.LE.250 )
$ SUBNAM( I: I ) = CHAR( IC-32 )
40 CONTINUE
END IF
END IF
*
C1 = SUBNAM( 1: 1 )
SNAME = C1.EQ.'S' .OR. C1.EQ.'D'
CNAME = C1.EQ.'C' .OR. C1.EQ.'Z'
IF( .NOT.( CNAME .OR. SNAME ) )
$ RETURN
C2 = SUBNAM( 2: 3 )
C3 = SUBNAM( 4: 6 )
C4 = C3( 2: 3 )
*
GO TO ( 50, 60, 70 )ISPEC
*
50 CONTINUE
*
* ISPEC = 1: block size
*
* In these examples, separate code is provided for setting NB for
* real and complex. We assume that NB will take the same value in
* single or double precision.
*
NB = 1
*
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR.
$ C3.EQ.'QLF' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NB = 32
ELSE
NB = 32
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'PO' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( SNAME .AND. C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRF' ) THEN
NB = 64
ELSE IF( C3.EQ.'TRD' ) THEN
NB = 32
ELSE IF( C3.EQ.'GST' ) THEN
NB = 64
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NB = 32
END IF
END IF
ELSE IF( C2.EQ.'GB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N4.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'PB' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
ELSE
IF( N2.LE.64 ) THEN
NB = 1
ELSE
NB = 32
END IF
END IF
END IF
ELSE IF( C2.EQ.'TR' ) THEN
IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( C2.EQ.'LA' ) THEN
IF( C3.EQ.'UUM' ) THEN
IF( SNAME ) THEN
NB = 64
ELSE
NB = 64
END IF
END IF
ELSE IF( SNAME .AND. C2.EQ.'ST' ) THEN
IF( C3.EQ.'EBZ' ) THEN
NB = 1
END IF
END IF
ILAENV = NB
RETURN
*
60 CONTINUE
*
* ISPEC = 2: minimum block size
*
NBMIN = 2
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
ELSE IF( C3.EQ.'TRI' ) THEN
IF( SNAME ) THEN
NBMIN = 2
ELSE
NBMIN = 2
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( C3.EQ.'TRF' ) THEN
IF( SNAME ) THEN
NBMIN = 8
ELSE
NBMIN = 8
END IF
ELSE IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NBMIN = 2
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
ELSE IF( C3( 1: 1 ).EQ.'M' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NBMIN = 2
END IF
END IF
END IF
ILAENV = NBMIN
RETURN
*
70 CONTINUE
*
* ISPEC = 3: crossover point
*
NX = 0
IF( C2.EQ.'GE' ) THEN
IF( C3.EQ.'QRF' .OR. C3.EQ.'RQF' .OR. C3.EQ.'LQF' .OR. C3.EQ.
$ 'QLF' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'HRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
ELSE IF( C3.EQ.'BRD' ) THEN
IF( SNAME ) THEN
NX = 128
ELSE
NX = 128
END IF
END IF
ELSE IF( C2.EQ.'SY' ) THEN
IF( SNAME .AND. C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( CNAME .AND. C2.EQ.'HE' ) THEN
IF( C3.EQ.'TRD' ) THEN
NX = 32
END IF
ELSE IF( SNAME .AND. C2.EQ.'OR' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
ELSE IF( CNAME .AND. C2.EQ.'UN' ) THEN
IF( C3( 1: 1 ).EQ.'G' ) THEN
IF( C4.EQ.'QR' .OR. C4.EQ.'RQ' .OR. C4.EQ.'LQ' .OR. C4.EQ.
$ 'QL' .OR. C4.EQ.'HR' .OR. C4.EQ.'TR' .OR. C4.EQ.'BR' )
$ THEN
NX = 128
END IF
END IF
END IF
ILAENV = NX
RETURN
*
80 CONTINUE
*
* ISPEC = 4: number of shifts (used by xHSEQR)
*
ILAENV = 6
RETURN
*
90 CONTINUE
*
* ISPEC = 5: minimum column dimension (not used)
*
ILAENV = 2
RETURN
*
100 CONTINUE
*
* ISPEC = 6: crossover point for SVD (used by xGELSS and xGESVD)
*
ILAENV = INT( REAL( MIN( N1, N2 ) )*1.6E0 )
RETURN
*
110 CONTINUE
*
* ISPEC = 7: number of processors (not used)
*
ILAENV = 1
RETURN
*
120 CONTINUE
*
* ISPEC = 8: crossover point for multishift (used by xHSEQR)
*
ILAENV = 50
RETURN
*
130 CONTINUE
*
* ISPEC = 9: maximum size of the subproblems at the bottom of the
* computation tree in the divide-and-conquer algorithm
* (used by xGELSD and xGESDD)
*
ILAENV = 25
RETURN
*
140 CONTINUE
*
* ISPEC = 10: ieee NaN arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 0, 0.0, 1.0 )
END IF
RETURN
*
150 CONTINUE
*
* ISPEC = 11: infinity arithmetic can be trusted not to trap
*
* ILAENV = 0
ILAENV = 1
IF( ILAENV.EQ.1 ) THEN
ILAENV = IEEECK( 1, 0.0, 1.0 )
END IF
RETURN
*
160 CONTINUE
*
* 12 <= ISPEC <= 16: xHSEQR or one of its subroutines.
*
ILAENV = IPARMQ( ISPEC, NAME, OPTS, N1, N2, N3, N4 )
RETURN
*
* End of ILAENV
*
END
INTEGER FUNCTION IEEECK( ISPEC, ZERO, ONE )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER ISPEC
REAL ONE, ZERO
* ..
*
* Purpose
* =======
*
* IEEECK is called from the ILAENV to verify that Infinity and
* possibly NaN arithmetic is safe (i.e. will not trap).
*
* Arguments
* =========
*
* ISPEC (input) INTEGER
* Specifies whether to test just for inifinity arithmetic
* or whether to test for infinity and NaN arithmetic.
* = 0: Verify infinity arithmetic only.
* = 1: Verify infinity and NaN arithmetic.
*
* ZERO (input) REAL
* Must contain the value 0.0
* This is passed to prevent the compiler from optimizing
* away this code.
*
* ONE (input) REAL
* Must contain the value 1.0
* This is passed to prevent the compiler from optimizing
* away this code.
*
* RETURN VALUE: INTEGER
* = 0: Arithmetic failed to produce the correct answers
* = 1: Arithmetic produced the correct answers
*
* .. Local Scalars ..
REAL NAN1, NAN2, NAN3, NAN4, NAN5, NAN6, NEGINF,
$ NEGZRO, NEWZRO, POSINF
* ..
* .. Executable Statements ..
IEEECK = 1
*
POSINF = ONE / ZERO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = -ONE / ZERO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGZRO = ONE / ( NEGINF+ONE )
IF( NEGZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = ONE / NEGZRO
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
NEWZRO = NEGZRO + ZERO
IF( NEWZRO.NE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = ONE / NEWZRO
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
NEGINF = NEGINF*POSINF
IF( NEGINF.GE.ZERO ) THEN
IEEECK = 0
RETURN
END IF
*
POSINF = POSINF*POSINF
IF( POSINF.LE.ONE ) THEN
IEEECK = 0
RETURN
END IF
*
*
*
*
* Return if we were only asked to check infinity arithmetic
*
IF( ISPEC.EQ.0 )
$ RETURN
*
NAN1 = POSINF + NEGINF
*
NAN2 = POSINF / NEGINF
*
NAN3 = POSINF / POSINF
*
NAN4 = POSINF*ZERO
*
NAN5 = NEGINF*NEGZRO
*
NAN6 = NAN5*0.0
*
IF( NAN1.EQ.NAN1 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN2.EQ.NAN2 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN3.EQ.NAN3 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN4.EQ.NAN4 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN5.EQ.NAN5 ) THEN
IEEECK = 0
RETURN
END IF
*
IF( NAN6.EQ.NAN6 ) THEN
IEEECK = 0
RETURN
END IF
*
RETURN
END
INTEGER FUNCTION IPARMQ( ISPEC, NAME, OPTS, N, ILO, IHI, LWORK )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, ISPEC, LWORK, N
CHARACTER NAME*( * ), OPTS*( * )
*
* Purpose
* =======
*
* This program sets problem and machine dependent parameters
* useful for xHSEQR and its subroutines. It is called whenever
* ILAENV is called with 12 <= ISPEC <= 16
*
* Arguments
* =========
*
* ISPEC (input) integer scalar
* ISPEC specifies which tunable parameter IPARMQ should
* return.
*
* ISPEC=12: (INMIN) Matrices of order nmin or less
* are sent directly to xLAHQR, the implicit
* double shift QR algorithm. NMIN must be
* at least 11.
*
* ISPEC=13: (INWIN) Size of the deflation window.
* This is best set greater than or equal to
* the number of simultaneous shifts NS.
* Larger matrices benefit from larger deflation
* windows.
*
* ISPEC=14: (INIBL) Determines when to stop nibbling and
* invest in an (expensive) multi-shift QR sweep.
* If the aggressive early deflation subroutine
* finds LD converged eigenvalues from an order
* NW deflation window and LD.GT.(NW*NIBBLE)/100,
* then the next QR sweep is skipped and early
* deflation is applied immediately to the
* remaining active diagonal block. Setting
* IPARMQ(ISPEC=14) = 0 causes TTQRE to skip a
* multi-shift QR sweep whenever early deflation
* finds a converged eigenvalue. Setting
* IPARMQ(ISPEC=14) greater than or equal to 100
* prevents TTQRE from skipping a multi-shift
* QR sweep.
*
* ISPEC=15: (NSHFTS) The number of simultaneous shifts in
* a multi-shift QR iteration.
*
* ISPEC=16: (IACC22) IPARMQ is set to 0, 1 or 2 with the
* following meanings.
* 0: During the multi-shift QR sweep,
* xLAQR5 does not accumulate reflections and
* does not use matrix-matrix multiply to
* update the far-from-diagonal matrix
* entries.
* 1: During the multi-shift QR sweep,
* xLAQR5 and/or xLAQRaccumulates reflections and uses
* matrix-matrix multiply to update the
* far-from-diagonal matrix entries.
* 2: During the multi-shift QR sweep.
* xLAQR5 accumulates reflections and takes
* advantage of 2-by-2 block structure during
* matrix-matrix multiplies.
* (If xTRMM is slower than xGEMM, then
* IPARMQ(ISPEC=16)=1 may be more efficient than
* IPARMQ(ISPEC=16)=2 despite the greater level of
* arithmetic work implied by the latter choice.)
*
* NAME (input) character string
* Name of the calling subroutine
*
* OPTS (input) character string
* This is a concatenation of the string arguments to
* TTQRE.
*
* N (input) integer scalar
* N is the order of the Hessenberg matrix H.
*
* ILO (input) INTEGER
* IHI (input) INTEGER
* It is assumed that H is already upper triangular
* in rows and columns 1:ILO-1 and IHI+1:N.
*
* LWORK (input) integer scalar
* The amount of workspace available.
*
* Further Details
* ===============
*
* Little is known about how best to choose these parameters.
* It is possible to use different values of the parameters
* for each of CHSEQR, DHSEQR, SHSEQR and ZHSEQR.
*
* It is probably best to choose different parameters for
* different matrices and different parameters at different
* times during the iteration, but this has not been
* implemented --- yet.
*
*
* The best choices of most of the parameters depend
* in an ill-understood way on the relative execution
* rate of xLAQR3 and xLAQR5 and on the nature of each
* particular eigenvalue problem. Experiment may be the
* only practical way to determine which choices are most
* effective.
*
* Following is a list of default values supplied by IPARMQ.
* These defaults may be adjusted in order to attain better
* performance in any particular computational environment.
*
* IPARMQ(ISPEC=12) The xLAHQR vs xLAQR0 crossover point.
* Default: 75. (Must be at least 11.)
*
* IPARMQ(ISPEC=13) Recommended deflation window size.
* This depends on ILO, IHI and NS, the
* number of simultaneous shifts returned
* by IPARMQ(ISPEC=15). The default for
* (IHI-ILO+1).LE.500 is NS. The default
* for (IHI-ILO+1).GT.500 is 3*NS/2.
*
* IPARMQ(ISPEC=14) Nibble crossover point. Default: 14.
*
* IPARMQ(ISPEC=15) Number of simultaneous shifts, NS.
* a multi-shift QR iteration.
*
* If IHI-ILO+1 is ...
*
* greater than ...but less ... the
* or equal to ... than default is
*
* 0 30 NS = 2+
* 30 60 NS = 4+
* 60 150 NS = 10
* 150 590 NS = **
* 590 3000 NS = 64
* 3000 6000 NS = 128
* 6000 infinity NS = 256
*
* (+) By default matrices of this order are
* passed to the implicit double shift routine
* xLAHQR. See IPARMQ(ISPEC=12) above. These
* values of NS are used only in case of a rare
* xLAHQR failure.
*
* (**) The asterisks (**) indicate an ad-hoc
* function increasing from 10 to 64.
*
* IPARMQ(ISPEC=16) Select structured matrix multiply.
* (See ISPEC=16 above for details.)
* Default: 3.
*
* ================================================================
* .. Parameters ..
INTEGER INMIN, INWIN, INIBL, ISHFTS, IACC22
PARAMETER ( INMIN = 12, INWIN = 13, INIBL = 14,
$ ISHFTS = 15, IACC22 = 16 )
INTEGER NMIN, K22MIN, KACMIN, NIBBLE, KNWSWP
PARAMETER ( NMIN = 75, K22MIN = 14, KACMIN = 14,
$ NIBBLE = 14, KNWSWP = 500 )
REAL TWO
PARAMETER ( TWO = 2.0 )
* ..
* .. Local Scalars ..
INTEGER NH, NS
* ..
* .. Intrinsic Functions ..
INTRINSIC LOG, MAX, MOD, NINT, REAL
* ..
* .. Executable Statements ..
IF( ( ISPEC.EQ.ISHFTS ) .OR. ( ISPEC.EQ.INWIN ) .OR.
$ ( ISPEC.EQ.IACC22 ) ) THEN
*
* ==== Set the number simultaneous shifts ====
*
NH = IHI - ILO + 1
NS = 2
IF( NH.GE.30 )
$ NS = 4
IF( NH.GE.60 )
$ NS = 10
IF( NH.GE.150 )
$ NS = MAX( 10, NH / NINT( LOG( REAL( NH ) ) / LOG( TWO ) ) )
IF( NH.GE.590 )
$ NS = 64
IF( NH.GE.3000 )
$ NS = 128
IF( NH.GE.6000 )
$ NS = 256
NS = MAX( 2, NS-MOD( NS, 2 ) )
END IF
*
IF( ISPEC.EQ.INMIN ) THEN
*
*
* ===== Matrices of order smaller than NMIN get sent
* . to xLAHQR, the classic double shift algorithm.
* . This must be at least 11. ====
*
IPARMQ = NMIN
*
ELSE IF( ISPEC.EQ.INIBL ) THEN
*
* ==== INIBL: skip a multi-shift qr iteration and
* . whenever aggressive early deflation finds
* . at least (NIBBLE*(window size)/100) deflations. ====
*
IPARMQ = NIBBLE
*
ELSE IF( ISPEC.EQ.ISHFTS ) THEN
*
* ==== NSHFTS: The number of simultaneous shifts =====
*
IPARMQ = NS
*
ELSE IF( ISPEC.EQ.INWIN ) THEN
*
* ==== NW: deflation window size. ====
*
IF( NH.LE.KNWSWP ) THEN
IPARMQ = NS
ELSE
IPARMQ = 3*NS / 2
END IF
*
ELSE IF( ISPEC.EQ.IACC22 ) THEN
*
* ==== IACC22: Whether to accumulate reflections
* . before updating the far-from-diagonal elements
* . and whether to use 2-by-2 block structure while
* . doing it. A small amount of work could be saved
* . by making this choice dependent also upon the
* . NH=IHI-ILO+1.
*
IPARMQ = 0
IF( NS.GE.KACMIN )
$ IPARMQ = 1
IF( NS.GE.K22MIN )
$ IPARMQ = 2
*
ELSE
* ===== invalid value of ispec =====
IPARMQ = -1
*
END IF
*
* ==== End of IPARMQ ====
*
END