DPTCON 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


 DPTTRF.


 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))).

N

          N is INTEGER


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

D

          D is DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


          factorization of A, as computed by DPTTRF.

E

          E is DOUBLE PRECISION 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 DPTTRF.

ANORM

          ANORM is DOUBLE PRECISION


          The 1-norm of the original matrix A.

RCOND

          RCOND is DOUBLE PRECISION


          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.

WORK

          WORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER


          = 0:  successful exit


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

  The method used is described in Nicholas J. Higham, 'Efficient


  Algorithms for Computing the Condition Number of a Tridiagonal


  Matrix', SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986.

 DPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using DPTTRF, and then calling DBDSQR 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 DSYTRD, DSPTRD, or DSBTRD 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.)

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.

N

          N is INTEGER


          The order of the matrix.  N >= 0.

D

          D is DOUBLE PRECISION array, dimension (N)


          On entry, the n diagonal elements of the tridiagonal


          matrix.


          On normal exit, D contains the eigenvalues, in descending


          order.

E

          E is DOUBLE PRECISION array, dimension (N-1)


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


          matrix.


          On exit, E has been destroyed.

Z

          Z is DOUBLE PRECISION 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.

LDZ

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          COMPZ = 'V' or 'I', LDZ >= max(1,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (4*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 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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DPTRFS 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.

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 DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the tridiagonal matrix A.

E

          E is DOUBLE PRECISION array, dimension (N-1)


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

DF

          DF is DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


          factorization computed by DPTTRF.

EF

          EF is DOUBLE PRECISION array, dimension (N-1)


          The (n-1) subdiagonal elements of the unit bidiagonal factor


          L from the factorization computed by DPTTRF.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          The right hand side matrix B.

LDB

          LDB is INTEGER


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

X

          X is DOUBLE PRECISION array, dimension (LDX,NRHS)


          On entry, the solution matrix X, as computed by DPTTRS.


          On exit, the improved solution matrix X.

LDX

          LDX is INTEGER


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

FERR

          FERR is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NRHS)


          The componentwise relative backward error of each solution


          vector X(j) (i.e., the smallest relative change in


          any element of A or B that makes X(j) an exact solution).

WORK

          WORK is DOUBLE PRECISION array, dimension (2*N)

INFO

          INFO is INTEGER


          = 0:  successful exit


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

  ITMAX is the maximum number of steps of iterative refinement.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DPTTRF computes the L*D*L**T factorization of a real symmetric


 positive definite tridiagonal matrix A.  The factorization may also


 be regarded as having the form A = U**T*D*U.

N

          N is INTEGER


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

D

          D is DOUBLE PRECISION 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 L*D*L**T factorization of A.

E

          E is DOUBLE PRECISION 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.

INFO

          INFO is INTEGER


          = 0: successful exit


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


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


               positive definite; if k < N, the factorization could not


               be completed, while if k = N, the factorization was


               completed, but D(N) <= 0.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DPTTRS solves a tridiagonal system of the form


    A * X = B


 using the L*D*L**T factorization of A computed by DPTTRF.  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.

N

          N is INTEGER


          The order of the tridiagonal 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 DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


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

E

          E is DOUBLE PRECISION 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.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          On entry, the right hand side vectors B for the system of


          linear equations.


          On exit, the solution vectors, 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 = -k, the k-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

 DPTTS2 solves a tridiagonal system of the form


    A * X = B


 using the L*D*L**T factorization of A computed by DPTTRF.  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.

N

          N is INTEGER


          The order of the tridiagonal 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 DOUBLE PRECISION array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


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

E

          E is DOUBLE PRECISION 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.

B

          B is DOUBLE PRECISION array, dimension (LDB,NRHS)


          On entry, the right hand side vectors B for the system of


          linear equations.


          On exit, the solution vectors, X.

LDB

          LDB is INTEGER


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.