LAPACK  3.4.2
LAPACK: Linear Algebra PACKage
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double
Computational routines
Collaboration diagram for double:

Functions/Subroutines

subroutine dptcon (N, D, E, ANORM, RCOND, WORK, INFO)
 DPTCON
subroutine dpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
 DPTEQR
subroutine dptrfs (N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, INFO)
 DPTRFS
subroutine dpttrf (N, D, E, INFO)
 DPTTRF
subroutine dpttrs (N, NRHS, D, E, B, LDB, INFO)
 DPTTRS
subroutine dptts2 (N, NRHS, D, E, B, LDB)
 DPTTS2 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 double computational functions for PT matrices

Function/Subroutine Documentation

subroutine dptcon ( integer  N,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision  ANORM,
double precision  RCOND,
double precision, dimension( * )  WORK,
integer  INFO 
)

DPTCON

Download DPTCON + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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))).
Parameters:
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in]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.
[in]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.
[in]ANORM
          ANORM is DOUBLE PRECISION
          The 1-norm of the original matrix A.
[out]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.
[out]WORK
          WORK is DOUBLE PRECISION 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.
Date:
September 2012
Further Details: 
  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.

Definition at line 119 of file dptcon.f.

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subroutine dpteqr ( character  COMPZ,
integer  N,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldz, * )  Z,
integer  LDZ,
double precision, dimension( * )  WORK,
integer  INFO 
)

DPTEQR

Download DPTEQR + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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.)
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 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.
[in,out]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.
[in,out]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.
[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 DOUBLE PRECISION 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.
Date:
September 2012

Definition at line 146 of file dpteqr.f.

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subroutine dptrfs ( integer  N,
integer  NRHS,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( * )  DF,
double precision, dimension( * )  EF,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldx, * )  X,
integer  LDX,
double precision, dimension( * )  FERR,
double precision, dimension( * )  BERR,
double precision, dimension( * )  WORK,
integer  INFO 
)

DPTRFS

Download DPTRFS + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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.
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 DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the tridiagonal matrix A.
[in]E
          E is DOUBLE PRECISION array, dimension (N-1)
          The (n-1) subdiagonal elements of the tridiagonal matrix A.
[in]DF
          DF is DOUBLE PRECISION array, dimension (N)
          The n diagonal elements of the diagonal matrix D from the
          factorization computed by DPTTRF.
[in]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.
[in]B
          B is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by DPTTRS.
          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 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).
[out]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).
[out]WORK
          WORK is DOUBLE PRECISION 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.
Date:
September 2012

Definition at line 163 of file dptrfs.f.

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subroutine dpttrf ( integer  N,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
integer  INFO 
)

DPTTRF

Download DPTTRF + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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.
Parameters:
[in]N
          N is INTEGER
          The order of the matrix A.  N >= 0.
[in,out]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.
[in,out]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.
[out]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.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012

Definition at line 92 of file dpttrf.f.

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subroutine dpttrs ( integer  N,
integer  NRHS,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldb, * )  B,
integer  LDB,
integer  INFO 
)

DPTTRS

Download DPTTRS + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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.
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 DOUBLE PRECISION 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 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.
[in,out]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.
[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.
Date:
September 2012

Definition at line 110 of file dpttrs.f.

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subroutine dptts2 ( integer  N,
integer  NRHS,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldb, * )  B,
integer  LDB 
)

DPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Download DPTTS2 + dependencies [TGZ] [ZIP] [TXT]
Purpose: 
 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.
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 DOUBLE PRECISION 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 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.
[in,out]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.
[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.
Date:
September 2012

Definition at line 103 of file dptts2.f.

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