SPTSV computes the solution to a real system of linear equations


 A*X = B, where A is an N-by-N symmetric positive definite tridiagonal


 matrix, and X and B are N-by-NRHS matrices.


 A is factored as A = L*D*L**T, and the factored form of A is then


 used to solve the system of equations.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrix B.  NRHS >= 0.

D

          D is REAL array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal matrix


          A.  On exit, the n diagonal elements of the diagonal matrix


          D from the factorization A = L*D*L**T.

E

          E is REAL array, dimension (N-1)


          On entry, the (n-1) subdiagonal elements of the tridiagonal


          matrix A.  On exit, 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 U**T*D*U factorization of A.)

B

          B is REAL array, dimension (LDB,NRHS)


          On entry, the N-by-NRHS right hand side matrix B.


          On exit, if INFO = 0, the N-by-NRHS solution matrix X.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value


          > 0:  if INFO = i, the leading minor of order i is not


                positive definite, and the solution has not been


                computed.  The factorization has not been completed


                unless i = N.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 SPTSVX uses the factorization A = L*D*L**T to compute the solution


 to a real system of linear equations A*X = B, where A is an N-by-N


 symmetric positive definite tridiagonal matrix and X and B are


 N-by-NRHS matrices.


 Error bounds on the solution and a condition estimate are also


 provided.

 The following steps are performed:


 1. If FACT = 'N', the matrix A is factored as A = L*D*L**T, where L


    is a unit lower bidiagonal matrix and D is diagonal.  The


    factorization can also be regarded as having the form


    A = U**T*D*U.


 2. If the leading i-by-i principal minor is not positive definite,


    then the routine returns with INFO = i. Otherwise, the factored


    form of A is used to estimate the condition number of the matrix


    A.  If the reciprocal of the condition number is less than machine


    precision, INFO = N+1 is returned as a warning, but the routine


    still goes on to solve for X and compute error bounds as


    described below.


 3. The system of equations is solved for X using the factored form


    of A.


 4. Iterative refinement is applied to improve the computed solution


    matrix and calculate error bounds and backward error estimates


    for it.

FACT

          FACT is CHARACTER*1


          Specifies whether or not the factored form of A has been


          supplied on entry.


          = 'F':  On entry, DF and EF contain the factored form of A.


                  D, E, DF, and EF will not be modified.


          = 'N':  The matrix A will be copied to DF and EF and


                  factored.

N

          N is INTEGER


          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

D

          D is REAL array, dimension (N)


          The n diagonal elements of the tridiagonal matrix A.

E

          E is REAL array, dimension (N-1)


          The (n-1) subdiagonal elements of the tridiagonal matrix A.

DF

          DF is REAL array, dimension (N)


          If FACT = 'F', then DF is an input argument and on entry


          contains the n diagonal elements of the diagonal matrix D


          from the L*D*L**T factorization of A.


          If FACT = 'N', then DF is an output argument and on exit


          contains the n diagonal elements of the diagonal matrix D


          from the L*D*L**T factorization of A.

EF

          EF is REAL array, dimension (N-1)


          If FACT = 'F', then EF is an input argument and on entry


          contains the (n-1) subdiagonal elements of the unit


          bidiagonal factor L from the L*D*L**T factorization of A.


          If FACT = 'N', then EF is an output argument and on exit


          contains the (n-1) subdiagonal elements of the unit


          bidiagonal factor L from the L*D*L**T factorization of A.

B

          B is REAL array, dimension (LDB,NRHS)


          The N-by-NRHS right hand side matrix B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is REAL array, dimension (LDX,NRHS)


          If INFO = 0 of INFO = N+1, the N-by-NRHS solution matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

RCOND

          RCOND is REAL


          The reciprocal condition number of the matrix A.  If RCOND


          is less than the machine precision (in particular, if


          RCOND = 0), the matrix is singular to working precision.


          This condition is indicated by a return code of INFO > 0.

FERR

          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).

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).

WORK

          WORK is REAL array, dimension (2*N)

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 leading minor of order i of A is


                       not positive definite, so the factorization


                       could not be completed, and the solution has not


                       been computed. RCOND = 0 is returned.


                = N+1: U is nonsingular, but RCOND is less than machine


                       precision, meaning that the matrix is singular


                       to working precision.  Nevertheless, the


                       solution and error bounds are computed because


                       there are a number of situations where the


                       computed solution can be more accurate than the


                       value of RCOND would suggest.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.