CPTCON computes the reciprocal of the condition number (in the


 1-norm) of a complex Hermitian positive definite tridiagonal matrix


 using the factorization A = L*D*L**H or A = U**H*D*U computed by


 CPTTRF.


 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 REAL array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


          factorization of A, as computed by CPTTRF.

E

          E is COMPLEX 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 CPTTRF.

ANORM

          ANORM is REAL


          The 1-norm of the original matrix A.

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.

RWORK

          RWORK is REAL 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.

 CPTEQR computes all eigenvalues and, optionally, eigenvectors of a


 symmetric positive definite tridiagonal matrix by first factoring the


 matrix using SPTTRF and then calling CBDSQR 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 positive definite Hermitian matrix


 can also be found if CHETRD, CHPTRD, or CHBTRD 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 Hermitian


                  matrix also.  Array Z contains the unitary 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 REAL 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 REAL array, dimension (N-1)


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


          matrix.


          On exit, E has been destroyed.

Z

          Z is COMPLEX array, dimension (LDZ, N)


          On entry, if COMPZ = 'V', the unitary matrix used in the


          reduction to tridiagonal form.


          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the


          original Hermitian 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 REAL 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.

 CPTRFS improves the computed solution to a system of linear


 equations when the coefficient matrix is Hermitian positive definite


 and tridiagonal, and provides error bounds and backward error


 estimates for the solution.

UPLO

          UPLO is CHARACTER*1


          Specifies whether the superdiagonal or the subdiagonal of the


          tridiagonal matrix A is stored and the form of the


          factorization:


          = 'U':  E is the superdiagonal of A, and A = U**H*D*U;


          = 'L':  E is the subdiagonal of A, and A = L*D*L**H.


          (The two forms are equivalent if A is real.)

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)


          The n real diagonal elements of the tridiagonal matrix A.

E

          E is COMPLEX array, dimension (N-1)


          The (n-1) off-diagonal elements of the tridiagonal matrix A


          (see UPLO).

DF

          DF is REAL array, dimension (N)


          The n diagonal elements of the diagonal matrix D from


          the factorization computed by CPTTRF.

EF

          EF is COMPLEX array, dimension (N-1)


          The (n-1) off-diagonal elements of the unit bidiagonal


          factor U or L from the factorization computed by CPTTRF


          (see UPLO).

B

          B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS)


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


          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 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 COMPLEX array, dimension (N)

RWORK

          RWORK is REAL array, dimension (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.

 CPTTRF computes the L*D*L**H factorization of a complex Hermitian


 positive definite tridiagonal matrix A.  The factorization may also


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

N

          N is INTEGER


          The order of the matrix A.  N >= 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 L*D*L**H factorization of A.

E

          E is COMPLEX 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**H factorization of A.


          E can also be regarded as the superdiagonal of the unit


          bidiagonal factor U from the U**H *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.

 CPTTRS solves a tridiagonal system of the form


    A * X = B


 using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF.


 D is a diagonal matrix specified in the vector D, U (or L) is a unit


 bidiagonal matrix whose superdiagonal (subdiagonal) is specified in


 the vector E, and X and B are N by NRHS matrices.

UPLO

          UPLO is CHARACTER*1


          Specifies the form of the factorization and whether the


          vector E is the superdiagonal of the upper bidiagonal factor


          U or the subdiagonal of the lower bidiagonal factor L.


          = 'U':  A = U**H*D*U, E is the superdiagonal of U


          = 'L':  A = L*D*L**H, E is the subdiagonal of L

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 REAL array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


          factorization A = U**H*D*U or A = L*D*L**H.

E

          E is COMPLEX array, dimension (N-1)


          If UPLO = 'U', the (n-1) superdiagonal elements of the unit


          bidiagonal factor U from the factorization A = U**H*D*U.


          If UPLO = 'L', the (n-1) subdiagonal elements of the unit


          bidiagonal factor L from the factorization A = L*D*L**H.

B

          B is COMPLEX 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.

 CPTTS2 solves a tridiagonal system of the form


    A * X = B


 using the factorization A = U**H*D*U or A = L*D*L**H computed by CPTTRF.


 D is a diagonal matrix specified in the vector D, U (or L) is a unit


 bidiagonal matrix whose superdiagonal (subdiagonal) is specified in


 the vector E, and X and B are N by NRHS matrices.

IUPLO

          IUPLO is INTEGER


          Specifies the form of the factorization and whether the


          vector E is the superdiagonal of the upper bidiagonal factor


          U or the subdiagonal of the lower bidiagonal factor L.


          = 1:  A = U**H *D*U, E is the superdiagonal of U


          = 0:  A = L*D*L**H, E is the subdiagonal of L

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 REAL array, dimension (N)


          The n diagonal elements of the diagonal matrix D from the


          factorization A = U**H *D*U or A = L*D*L**H.

E

          E is COMPLEX array, dimension (N-1)


          If IUPLO = 1, the (n-1) superdiagonal elements of the unit


          bidiagonal factor U from the factorization A = U**H*D*U.


          If IUPLO = 0, the (n-1) subdiagonal elements of the unit


          bidiagonal factor L from the factorization A = L*D*L**H.

B

          B is COMPLEX 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.