real

Functions
subroutine	sptcon (n, d, e, anorm, rcond, work, info)
	SPTCON
subroutine	spteqr (compz, n, d, e, z, ldz, work, info)
	SPTEQR
subroutine	sptrfs (n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
	SPTRFS
subroutine	spttrs (n, nrhs, d, e, b, ldb, info)
	SPTTRS
subroutine	sptts2 (n, nrhs, d, e, b, ldb)
	SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Detailed Description

This is the group of real computational functions for PT matrices

Function Documentation

sptcon()

subroutine sptcon	(	integer	n,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real	anorm,
		real	rcond,
		real, dimension( * )	work,
		integer	info )

SPTCON

Download SPTCON + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SPTCON computes the reciprocal of the condition number (in the
!> 1-norm) of a real symmetric positive definite tridiagonal matrix
!> using the factorization A = L*D*L**T or A = U**T*D*U computed by
!> SPTTRF.
!>
!> Norm(inv(A)) is computed by a direct method, and the reciprocal of
!> the condition number is computed as
!>              RCOND = 1 / (ANORM * norm(inv(A))).
!> 

Parameters

[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the diagonal matrix D from the !> factorization of A, as computed by SPTTRF. !>
[in]	E	!> E is REAL array, dimension (N-1) !> The (n-1) off-diagonal elements of the unit bidiagonal factor !> U or L from the factorization of A, as computed by SPTTRF. !>
[in]	ANORM	!> ANORM is REAL !> The 1-norm of the original matrix A. !>
[out]	RCOND	!> RCOND is REAL !> The reciprocal of the condition number of the matrix A, !> computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the !> 1-norm of inv(A) computed in this routine. !>
[out]	WORK	!> WORK is REAL array, dimension (N) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!>
!>  The method used is described in Nicholas J. Higham, , SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.
!> 

Definition at line 117 of file sptcon.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, N
      REAL               ANORM, RCOND
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ONE, ZERO
      parameter( one = 1.0e+0, zero = 0.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IX
      REAL               AINVNM
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      EXTERNAL           isamax
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      info = 0
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( anorm.LT.zero ) THEN
         info = -4
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTCON', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      rcond = zero
      IF( n.EQ.0 ) THEN
         rcond = one
         RETURN
      ELSE IF( anorm.EQ.zero ) THEN
         RETURN
      END IF
*
*     Check that D(1:N) is positive.
*
      DO 10 i = 1, n
         IF( d( i ).LE.zero )
     $      RETURN
   10 CONTINUE
*
*     Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
*        m(i,j) =  abs(A(i,j)), i = j,
*        m(i,j) = -abs(A(i,j)), i .ne. j,
*
*     and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**T.
*
*     Solve M(L) * x = e.
*
      work( 1 ) = one
      DO 20 i = 2, n
         work( i ) = one + work( i-1 )*abs( e( i-1 ) )
   20 CONTINUE
*
*     Solve D * M(L)**T * x = b.
*
      work( n ) = work( n ) / d( n )
      DO 30 i = n - 1, 1, -1
         work( i ) = work( i ) / d( i ) + work( i+1 )*abs( e( i ) )
   30 CONTINUE
*
*     Compute AINVNM = max(x(i)), 1<=i<=n.
*
      ix = isamax( n, work, 1 )
      ainvnm = abs( work( ix ) )
*
*     Compute the reciprocal condition number.
*
      IF( ainvnm.NE.zero )
     $   rcond = ( one / ainvnm ) / anorm
*
      RETURN
*
*     End of SPTCON
*

spteqr()

subroutine spteqr	(	character	compz,
		integer	n,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( ldz, * )	z,
		integer	ldz,
		real, dimension( * )	work,
		integer	info )

SPTEQR

Download SPTEQR + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
!> symmetric positive definite tridiagonal matrix by first factoring the
!> matrix using SPTTRF, and then calling SBDSQR to compute the singular
!> values of the bidiagonal factor.
!>
!> This routine computes the eigenvalues of the positive definite
!> tridiagonal matrix to high relative accuracy.  This means that if the
!> eigenvalues range over many orders of magnitude in size, then the
!> small eigenvalues and corresponding eigenvectors will be computed
!> more accurately than, for example, with the standard QR method.
!>
!> The eigenvectors of a full or band symmetric positive definite matrix
!> can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
!> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
!> form, however, may preclude the possibility of obtaining high
!> relative accuracy in the small eigenvalues of the original matrix, if
!> these eigenvalues range over many orders of magnitude.)
!> 

Parameters

[in]	COMPZ	!> COMPZ is CHARACTER*1 !> = 'N': Compute eigenvalues only. !> = 'V': Compute eigenvectors of original symmetric !> matrix also. Array Z contains the orthogonal !> matrix used to reduce the original matrix to !> tridiagonal form. !> = 'I': Compute eigenvectors of tridiagonal matrix also. !>
[in]	N	!> N is INTEGER !> The order of the matrix. N >= 0. !>
[in,out]	D	!> D is REAL array, dimension (N) !> On entry, the n diagonal elements of the tridiagonal !> matrix. !> On normal exit, D contains the eigenvalues, in descending !> order. !>
[in,out]	E	!> E is REAL array, dimension (N-1) !> On entry, the (n-1) subdiagonal elements of the tridiagonal !> matrix. !> On exit, E has been destroyed. !>
[in,out]	Z	!> Z is REAL array, dimension (LDZ, N) !> On entry, if COMPZ = 'V', the orthogonal matrix used in the !> reduction to tridiagonal form. !> On exit, if COMPZ = 'V', the orthonormal eigenvectors of the !> original symmetric matrix; !> if COMPZ = 'I', the orthonormal eigenvectors of the !> tridiagonal matrix. !> If INFO > 0 on exit, Z contains the eigenvectors associated !> with only the stored eigenvalues. !> If COMPZ = 'N', then Z is not referenced. !>
[in]	LDZ	!> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1, and if !> COMPZ = 'V' or 'I', LDZ >= max(1,N). !>
[out]	WORK	!> WORK is REAL array, dimension (4*N) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !> > 0: if INFO = i, and i is: !> <= N the Cholesky factorization of the matrix could !> not be performed because the i-th principal minor !> was not positive definite. !> > N the SVD algorithm failed to converge; !> if INFO = N+i, i off-diagonal elements of the !> bidiagonal factor did not converge to zero. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 144 of file spteqr.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPZ
      INTEGER            INFO, LDZ, N
*     ..
*     .. Array Arguments ..
      REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           sbdsqr, slaset, spttrf, xerbla
*     ..
*     .. Local Arrays ..
      REAL               C( 1, 1 ), VT( 1, 1 )
*     ..
*     .. Local Scalars ..
      INTEGER            I, ICOMPZ, NRU
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( lsame( compz, 'N' ) ) THEN
         icompz = 0
      ELSE IF( lsame( compz, 'V' ) ) THEN
         icompz = 1
      ELSE IF( lsame( compz, 'I' ) ) THEN
         icompz = 2
      ELSE
         icompz = -1
      END IF
      IF( icompz.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
     $         n ) ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTEQR', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
      IF( n.EQ.1 ) THEN
         IF( icompz.GT.0 )
     $      z( 1, 1 ) = one
         RETURN
      END IF
      IF( icompz.EQ.2 )
     $   CALL slaset( 'Full', n, n, zero, one, z, ldz )
*
*     Call SPTTRF to factor the matrix.
*
      CALL spttrf( n, d, e, info )
      IF( info.NE.0 )
     $   RETURN
      DO 10 i = 1, n
         d( i ) = sqrt( d( i ) )
   10 CONTINUE
      DO 20 i = 1, n - 1
         e( i ) = e( i )*d( i )
   20 CONTINUE
*
*     Call SBDSQR to compute the singular values/vectors of the
*     bidiagonal factor.
*
      IF( icompz.GT.0 ) THEN
         nru = n
      ELSE
         nru = 0
      END IF
      CALL sbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
     $             work, info )
*
*     Square the singular values.
*
      IF( info.EQ.0 ) THEN
         DO 30 i = 1, n
            d( i ) = d( i )*d( i )
   30    CONTINUE
      ELSE
         info = n + info
      END IF
*
      RETURN
*
*     End of SPTEQR
*

sptrfs()

subroutine sptrfs	(	integer	n,
		integer	nrhs,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( * )	df,
		real, dimension( * )	ef,
		real, dimension( ldb, * )	b,
		integer	ldb,
		real, dimension( ldx, * )	x,
		integer	ldx,
		real, dimension( * )	ferr,
		real, dimension( * )	berr,
		real, dimension( * )	work,
		integer	info )

SPTRFS

Download SPTRFS + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SPTRFS improves the computed solution to a system of linear
!> equations when the coefficient matrix is symmetric positive definite
!> and tridiagonal, and provides error bounds and backward error
!> estimates for the solution.
!> 

Parameters

[in]	N	!> N is INTEGER !> The order of the matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the tridiagonal matrix A. !>
[in]	E	!> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the tridiagonal matrix A. !>
[in]	DF	!> DF is REAL array, dimension (N) !> The n diagonal elements of the diagonal matrix D from the !> factorization computed by SPTTRF. !>
[in]	EF	!> EF is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the unit bidiagonal factor !> L from the factorization computed by SPTTRF. !>
[in]	B	!> B is REAL array, dimension (LDB,NRHS) !> The right hand side matrix B. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[in,out]	X	!> X is REAL array, dimension (LDX,NRHS) !> On entry, the solution matrix X, as computed by SPTTRS. !> On exit, the improved solution matrix X. !>
[in]	LDX	!> LDX is INTEGER !> The leading dimension of the array X. LDX >= max(1,N). !>
[out]	FERR	!> FERR is REAL array, dimension (NRHS) !> The forward error bound for each solution vector !> X(j) (the j-th column of the solution matrix X). !> If XTRUE is the true solution corresponding to X(j), FERR(j) !> is an estimated upper bound for the magnitude of the largest !> element in (X(j) - XTRUE) divided by the magnitude of the !> largest element in X(j). !>
[out]	BERR	!> BERR is REAL array, dimension (NRHS) !> The componentwise relative backward error of each solution !> vector X(j) (i.e., the smallest relative change in !> any element of A or B that makes X(j) an exact solution). !>
[out]	WORK	!> WORK is REAL array, dimension (2*N) !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !>

Internal Parameters:

!>  ITMAX is the maximum number of steps of iterative refinement.
!> 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 161 of file sptrfs.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, LDX, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               B( LDB, * ), BERR( * ), D( * ), DF( * ),
     $                   E( * ), EF( * ), FERR( * ), WORK( * ),
     $                   X( LDX, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            ITMAX
      parameter( itmax = 5 )
      REAL               ZERO
      parameter( zero = 0.0e+0 )
      REAL               ONE
      parameter( one = 1.0e+0 )
      REAL               TWO
      parameter( two = 2.0e+0 )
      REAL               THREE
      parameter( three = 3.0e+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            COUNT, I, IX, J, NZ
      REAL               BI, CX, DX, EPS, EX, LSTRES, S, SAFE1, SAFE2,
     $                   SAFMIN
*     ..
*     .. External Subroutines ..
      EXTERNAL           saxpy, spttrs, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max
*     ..
*     .. External Functions ..
      INTEGER            ISAMAX
      REAL               SLAMCH
      EXTERNAL           isamax, slamch
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( nrhs.LT.0 ) THEN
         info = -2
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -8
      ELSE IF( ldx.LT.max( 1, n ) ) THEN
         info = -10
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTRFS', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
         DO 10 j = 1, nrhs
            ferr( j ) = zero
            berr( j ) = zero
   10    CONTINUE
         RETURN
      END IF
*
*     NZ = maximum number of nonzero elements in each row of A, plus 1
*
      nz = 4
      eps = slamch( 'Epsilon' )
      safmin = slamch( 'Safe minimum' )
      safe1 = nz*safmin
      safe2 = safe1 / eps
*
*     Do for each right hand side
*
      DO 90 j = 1, nrhs
*
         count = 1
         lstres = three
   20    CONTINUE
*
*        Loop until stopping criterion is satisfied.
*
*        Compute residual R = B - A * X.  Also compute
*        abs(A)*abs(x) + abs(b) for use in the backward error bound.
*
         IF( n.EQ.1 ) THEN
            bi = b( 1, j )
            dx = d( 1 )*x( 1, j )
            work( n+1 ) = bi - dx
            work( 1 ) = abs( bi ) + abs( dx )
         ELSE
            bi = b( 1, j )
            dx = d( 1 )*x( 1, j )
            ex = e( 1 )*x( 2, j )
            work( n+1 ) = bi - dx - ex
            work( 1 ) = abs( bi ) + abs( dx ) + abs( ex )
            DO 30 i = 2, n - 1
               bi = b( i, j )
               cx = e( i-1 )*x( i-1, j )
               dx = d( i )*x( i, j )
               ex = e( i )*x( i+1, j )
               work( n+i ) = bi - cx - dx - ex
               work( i ) = abs( bi ) + abs( cx ) + abs( dx ) + abs( ex )
   30       CONTINUE
            bi = b( n, j )
            cx = e( n-1 )*x( n-1, j )
            dx = d( n )*x( n, j )
            work( n+n ) = bi - cx - dx
            work( n ) = abs( bi ) + abs( cx ) + abs( dx )
         END IF
*
*        Compute componentwise relative backward error from formula
*
*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
*
*        where abs(Z) is the componentwise absolute value of the matrix
*        or vector Z.  If the i-th component of the denominator is less
*        than SAFE2, then SAFE1 is added to the i-th components of the
*        numerator and denominator before dividing.
*
         s = zero
         DO 40 i = 1, n
            IF( work( i ).GT.safe2 ) THEN
               s = max( s, abs( work( n+i ) ) / work( i ) )
            ELSE
               s = max( s, ( abs( work( n+i ) )+safe1 ) /
     $             ( work( i )+safe1 ) )
            END IF
   40    CONTINUE
         berr( j ) = s
*
*        Test stopping criterion. Continue iterating if
*           1) The residual BERR(J) is larger than machine epsilon, and
*           2) BERR(J) decreased by at least a factor of 2 during the
*              last iteration, and
*           3) At most ITMAX iterations tried.
*
         IF( berr( j ).GT.eps .AND. two*berr( j ).LE.lstres .AND.
     $       count.LE.itmax ) THEN
*
*           Update solution and try again.
*
            CALL spttrs( n, 1, df, ef, work( n+1 ), n, info )
            CALL saxpy( n, one, work( n+1 ), 1, x( 1, j ), 1 )
            lstres = berr( j )
            count = count + 1
            GO TO 20
         END IF
*
*        Bound error from formula
*
*        norm(X - XTRUE) / norm(X) .le. FERR =
*        norm( abs(inv(A))*
*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
*
*        where
*          norm(Z) is the magnitude of the largest component of Z
*          inv(A) is the inverse of A
*          abs(Z) is the componentwise absolute value of the matrix or
*             vector Z
*          NZ is the maximum number of nonzeros in any row of A, plus 1
*          EPS is machine epsilon
*
*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
*        is incremented by SAFE1 if the i-th component of
*        abs(A)*abs(X) + abs(B) is less than SAFE2.
*
         DO 50 i = 1, n
            IF( work( i ).GT.safe2 ) THEN
               work( i ) = abs( work( n+i ) ) + nz*eps*work( i )
            ELSE
               work( i ) = abs( work( n+i ) ) + nz*eps*work( i ) + safe1
            END IF
   50    CONTINUE
         ix = isamax( n, work, 1 )
         ferr( j ) = work( ix )
*
*        Estimate the norm of inv(A).
*
*        Solve M(A) * x = e, where M(A) = (m(i,j)) is given by
*
*           m(i,j) =  abs(A(i,j)), i = j,
*           m(i,j) = -abs(A(i,j)), i .ne. j,
*
*        and e = [ 1, 1, ..., 1 ]**T.  Note M(A) = M(L)*D*M(L)**T.
*
*        Solve M(L) * x = e.
*
         work( 1 ) = one
         DO 60 i = 2, n
            work( i ) = one + work( i-1 )*abs( ef( i-1 ) )
   60    CONTINUE
*
*        Solve D * M(L)**T * x = b.
*
         work( n ) = work( n ) / df( n )
         DO 70 i = n - 1, 1, -1
            work( i ) = work( i ) / df( i ) + work( i+1 )*abs( ef( i ) )
   70    CONTINUE
*
*        Compute norm(inv(A)) = max(x(i)), 1<=i<=n.
*
         ix = isamax( n, work, 1 )
         ferr( j ) = ferr( j )*abs( work( ix ) )
*
*        Normalize error.
*
         lstres = zero
         DO 80 i = 1, n
            lstres = max( lstres, abs( x( i, j ) ) )
   80    CONTINUE
         IF( lstres.NE.zero )
     $      ferr( j ) = ferr( j ) / lstres
*
   90 CONTINUE
*
      RETURN
*
*     End of SPTRFS
*

spttrs()

subroutine spttrs	(	integer	n,
		integer	nrhs,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( ldb, * )	b,
		integer	ldb,
		integer	info )

SPTTRS

Download SPTTRS + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SPTTRS solves a tridiagonal system of the form
!>    A * X = B
!> using the L*D*L**T factorization of A computed by SPTTRF.  D is a
!> diagonal matrix specified in the vector D, L is a unit bidiagonal
!> matrix whose subdiagonal is specified in the vector E, and X and B
!> are N by NRHS matrices.
!> 

Parameters

[in]	N	!> N is INTEGER !> The order of the tridiagonal matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the diagonal matrix D from the !> L*D*L**T factorization of A. !>
[in]	E	!> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the unit bidiagonal factor !> L from the L*D*L**T factorization of A. E can also be regarded !> as the superdiagonal of the unit bidiagonal factor U from the !> factorization A = U**T*D*U. !>
[in,out]	B	!> B is REAL array, dimension (LDB,NRHS) !> On entry, the right hand side vectors B for the system of !> linear equations. !> On exit, the solution vectors, X. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 108 of file spttrs.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               B( LDB, * ), D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            J, JB, NB
*     ..
*     .. External Functions ..
      INTEGER            ILAENV
      EXTERNAL           ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL           sptts2, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, min
*     ..
*     .. Executable Statements ..
*
*     Test the input arguments.
*
      info = 0
      IF( n.LT.0 ) THEN
         info = -1
      ELSE IF( nrhs.LT.0 ) THEN
         info = -2
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTTRS', -info )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 .OR. nrhs.EQ.0 )
     $   RETURN
*
*     Determine the number of right-hand sides to solve at a time.
*
      IF( nrhs.EQ.1 ) THEN
         nb = 1
      ELSE
         nb = max( 1, ilaenv( 1, 'SPTTRS', ' ', n, nrhs, -1, -1 ) )
      END IF
*
      IF( nb.GE.nrhs ) THEN
         CALL sptts2( n, nrhs, d, e, b, ldb )
      ELSE
         DO 10 j = 1, nrhs, nb
            jb = min( nrhs-j+1, nb )
            CALL sptts2( n, jb, d, e, b( 1, j ), ldb )
   10    CONTINUE
      END IF
*
      RETURN
*
*     End of SPTTRS
*

sptts2()

subroutine sptts2	(	integer	n,
		integer	nrhs,
		real, dimension( * )	d,
		real, dimension( * )	e,
		real, dimension( ldb, * )	b,
		integer	ldb )

SPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Download SPTTS2 + dependencies [TGZ] [ZIP] [TXT]

Purpose:

!>
!> SPTTS2 solves a tridiagonal system of the form
!>    A * X = B
!> using the L*D*L**T factorization of A computed by SPTTRF.  D is a
!> diagonal matrix specified in the vector D, L is a unit bidiagonal
!> matrix whose subdiagonal is specified in the vector E, and X and B
!> are N by NRHS matrices.
!> 

Parameters

[in]	N	!> N is INTEGER !> The order of the tridiagonal matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !>
[in]	D	!> D is REAL array, dimension (N) !> The n diagonal elements of the diagonal matrix D from the !> L*D*L**T factorization of A. !>
[in]	E	!> E is REAL array, dimension (N-1) !> The (n-1) subdiagonal elements of the unit bidiagonal factor !> L from the L*D*L**T factorization of A. E can also be regarded !> as the superdiagonal of the unit bidiagonal factor U from the !> factorization A = U**T*D*U. !>
[in,out]	B	!> B is REAL array, dimension (LDB,NRHS) !> On entry, the right hand side vectors B for the system of !> linear equations. !> On exit, the solution vectors, X. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 101 of file sptts2.f.

*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            LDB, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               B( LDB, * ), D( * ), E( * )
*     ..
*
*  =====================================================================
*
*     .. Local Scalars ..
      INTEGER            I, J
*     ..
*     .. External Subroutines ..
      EXTERNAL           sscal
*     ..
*     .. Executable Statements ..
*
*     Quick return if possible
*
      IF( n.LE.1 ) THEN
         IF( n.EQ.1 )
     $      CALL sscal( nrhs, 1. / d( 1 ), b, ldb )
         RETURN
      END IF
*
*     Solve A * X = B using the factorization A = L*D*L**T,
*     overwriting each right hand side vector with its solution.
*
      DO 30 j = 1, nrhs
*
*           Solve L * x = b.
*
         DO 10 i = 2, n
            b( i, j ) = b( i, j ) - b( i-1, j )*e( i-1 )
   10    CONTINUE
*
*           Solve D * L**T * x = b.
*
         b( n, j ) = b( n, j ) / d( n )
         DO 20 i = n - 1, 1, -1
            b( i, j ) = b( i, j ) / d( i ) - b( i+1, j )*e( i )
   20    CONTINUE
   30 CONTINUE
*
      RETURN
*
*     End of SPTTS2
*