*> \brief \b SLAGTS solves the system of equations (T-λI)x = y or (T-λI)Tx = y,where T is a general tridiagonal matrix and λ a scalar, using the LU factorization computed by slagtf.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SLAGTS + dependencies
*>
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*>
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*
* Definition:
* ===========
*
* SUBROUTINE SLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, JOB, N
* REAL TOL
* ..
* .. Array Arguments ..
* INTEGER IN( * )
* REAL A( * ), B( * ), C( * ), D( * ), Y( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLAGTS may be used to solve one of the systems of equations
*>
*> (T - lambda*I)*x = y or (T - lambda*I)**T*x = y,
*>
*> where T is an n by n tridiagonal matrix, for x, following the
*> factorization of (T - lambda*I) as
*>
*> (T - lambda*I) = P*L*U ,
*>
*> by routine SLAGTF. The choice of equation to be solved is
*> controlled by the argument JOB, and in each case there is an option
*> to perturb zero or very small diagonal elements of U, this option
*> being intended for use in applications such as inverse iteration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is INTEGER
*> Specifies the job to be performed by SLAGTS as follows:
*> = 1: The equations (T - lambda*I)x = y are to be solved,
*> but diagonal elements of U are not to be perturbed.
*> = -1: The equations (T - lambda*I)x = y are to be solved
*> and, if overflow would otherwise occur, the diagonal
*> elements of U are to be perturbed. See argument TOL
*> below.
*> = 2: The equations (T - lambda*I)**Tx = y are to be solved,
*> but diagonal elements of U are not to be perturbed.
*> = -2: The equations (T - lambda*I)**Tx = y are to be solved
*> and, if overflow would otherwise occur, the diagonal
*> elements of U are to be perturbed. See argument TOL
*> below.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix T.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is REAL array, dimension (N)
*> On entry, A must contain the diagonal elements of U as
*> returned from SLAGTF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is REAL array, dimension (N-1)
*> On entry, B must contain the first super-diagonal elements of
*> U as returned from SLAGTF.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is REAL array, dimension (N-1)
*> On entry, C must contain the sub-diagonal elements of L as
*> returned from SLAGTF.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*> D is REAL array, dimension (N-2)
*> On entry, D must contain the second super-diagonal elements
*> of U as returned from SLAGTF.
*> \endverbatim
*>
*> \param[in] IN
*> \verbatim
*> IN is INTEGER array, dimension (N)
*> On entry, IN must contain details of the matrix P as returned
*> from SLAGTF.
*> \endverbatim
*>
*> \param[in,out] Y
*> \verbatim
*> Y is REAL array, dimension (N)
*> On entry, the right hand side vector y.
*> On exit, Y is overwritten by the solution vector x.
*> \endverbatim
*>
*> \param[in,out] TOL
*> \verbatim
*> TOL is REAL
*> On entry, with JOB .lt. 0, TOL should be the minimum
*> perturbation to be made to very small diagonal elements of U.
*> TOL should normally be chosen as about eps*norm(U), where eps
*> is the relative machine precision, but if TOL is supplied as
*> non-positive, then it is reset to eps*max( abs( u(i,j) ) ).
*> If JOB .gt. 0 then TOL is not referenced.
*>
*> On exit, TOL is changed as described above, only if TOL is
*> non-positive on entry. Otherwise TOL is unchanged.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0 : successful exit
*> .lt. 0: if INFO = -i, the i-th argument had an illegal value
*> .gt. 0: overflow would occur when computing the INFO(th)
*> element of the solution vector x. This can only occur
*> when JOB is supplied as positive and either means
*> that a diagonal element of U is very small, or that
*> the elements of the right-hand side vector y are very
*> large.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup OTHERauxiliary
*
* =====================================================================
SUBROUTINE SLAGTS( JOB, N, A, B, C, D, IN, Y, TOL, INFO )
*
* -- LAPACK auxiliary routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
INTEGER INFO, JOB, N
REAL TOL
* ..
* .. Array Arguments ..
INTEGER IN( * )
REAL A( * ), B( * ), C( * ), D( * ), Y( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE, ZERO
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
* ..
* .. Local Scalars ..
INTEGER K
REAL ABSAK, AK, BIGNUM, EPS, PERT, SFMIN, TEMP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SIGN
* ..
* .. External Functions ..
REAL SLAMCH
EXTERNAL SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL XERBLA
* ..
* .. Executable Statements ..
*
INFO = 0
IF( ( ABS( JOB ).GT.2 ) .OR. ( JOB.EQ.0 ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLAGTS', -INFO )
RETURN
END IF
*
IF( N.EQ.0 )
$ RETURN
*
EPS = SLAMCH( 'Epsilon' )
SFMIN = SLAMCH( 'Safe minimum' )
BIGNUM = ONE / SFMIN
*
IF( JOB.LT.0 ) THEN
IF( TOL.LE.ZERO ) THEN
TOL = ABS( A( 1 ) )
IF( N.GT.1 )
$ TOL = MAX( TOL, ABS( A( 2 ) ), ABS( B( 1 ) ) )
DO 10 K = 3, N
TOL = MAX( TOL, ABS( A( K ) ), ABS( B( K-1 ) ),
$ ABS( D( K-2 ) ) )
10 CONTINUE
TOL = TOL*EPS
IF( TOL.EQ.ZERO )
$ TOL = EPS
END IF
END IF
*
IF( ABS( JOB ).EQ.1 ) THEN
DO 20 K = 2, N
IF( IN( K-1 ).EQ.0 ) THEN
Y( K ) = Y( K ) - C( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K-1 )
Y( K-1 ) = Y( K )
Y( K ) = TEMP - C( K-1 )*Y( K )
END IF
20 CONTINUE
IF( JOB.EQ.1 ) THEN
DO 30 K = N, 1, -1
IF( K.LE.N-2 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
ELSE IF( K.EQ.N-1 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
INFO = K
RETURN
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
INFO = K
RETURN
END IF
END IF
Y( K ) = TEMP / AK
30 CONTINUE
ELSE
DO 50 K = N, 1, -1
IF( K.LE.N-2 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 ) - D( K )*Y( K+2 )
ELSE IF( K.EQ.N-1 ) THEN
TEMP = Y( K ) - B( K )*Y( K+1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
PERT = SIGN( TOL, AK )
40 CONTINUE
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 40
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 40
END IF
END IF
Y( K ) = TEMP / AK
50 CONTINUE
END IF
ELSE
*
* Come to here if JOB = 2 or -2
*
IF( JOB.EQ.2 ) THEN
DO 60 K = 1, N
IF( K.GE.3 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
ELSE IF( K.EQ.2 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
INFO = K
RETURN
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
INFO = K
RETURN
END IF
END IF
Y( K ) = TEMP / AK
60 CONTINUE
ELSE
DO 80 K = 1, N
IF( K.GE.3 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 ) - D( K-2 )*Y( K-2 )
ELSE IF( K.EQ.2 ) THEN
TEMP = Y( K ) - B( K-1 )*Y( K-1 )
ELSE
TEMP = Y( K )
END IF
AK = A( K )
PERT = SIGN( TOL, AK )
70 CONTINUE
ABSAK = ABS( AK )
IF( ABSAK.LT.ONE ) THEN
IF( ABSAK.LT.SFMIN ) THEN
IF( ABSAK.EQ.ZERO .OR. ABS( TEMP )*SFMIN.GT.ABSAK )
$ THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 70
ELSE
TEMP = TEMP*BIGNUM
AK = AK*BIGNUM
END IF
ELSE IF( ABS( TEMP ).GT.ABSAK*BIGNUM ) THEN
AK = AK + PERT
PERT = 2*PERT
GO TO 70
END IF
END IF
Y( K ) = TEMP / AK
80 CONTINUE
END IF
*
DO 90 K = N, 2, -1
IF( IN( K-1 ).EQ.0 ) THEN
Y( K-1 ) = Y( K-1 ) - C( K-1 )*Y( K )
ELSE
TEMP = Y( K-1 )
Y( K-1 ) = Y( K )
Y( K ) = TEMP - C( K-1 )*Y( K )
END IF
90 CONTINUE
END IF
*
* End of SLAGTS
*
END