LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
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## Functions/Subroutines

subroutine cptcon (N, D, E, ANORM, RCOND, RWORK, INFO)
CPTCON
subroutine cpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
CPTEQR
subroutine cptrfs (UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)
CPTRFS
subroutine cpttrf (N, D, E, INFO)
CPTTRF
subroutine cpttrs (UPLO, N, NRHS, D, E, B, LDB, INFO)
CPTTRS
subroutine cptts2 (IUPLO, N, NRHS, D, E, B, LDB)
CPTTS2 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 complex computational functions for PT matrices

## Function/Subroutine Documentation

 subroutine cptcon ( integer N, real, dimension( * ) D, complex, dimension( * ) E, real ANORM, real RCOND, real, dimension( * ) RWORK, integer INFO )

CPTCON

Purpose:
``` 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))).```
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 CPTTRF.``` [in] 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.``` [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] RWORK ` RWORK 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```
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 120 of file cptcon.f.

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 subroutine cpteqr ( character COMPZ, integer N, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer INFO )

CPTEQR

Purpose:
``` 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.)```
Parameters:
 [in] 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.``` [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 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.``` [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.```
Date:
September 2012

Definition at line 146 of file cpteqr.f.

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 subroutine cptrfs ( character UPLO, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, real, dimension( * ) DF, complex, dimension( * ) EF, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer INFO )

CPTRFS

Purpose:
``` 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.```
Parameters:
 [in] 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.)``` [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 real diagonal elements of the tridiagonal matrix A.``` [in] E ``` E is COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix A (see UPLO).``` [in] DF ``` DF is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by CPTTRF.``` [in] 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).``` [in] B ``` B is COMPLEX 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 COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CPTTRS. 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 COMPLEX array, dimension (N)` [out] RWORK ` RWORK 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```
Internal Parameters:
`  ITMAX is the maximum number of steps of iterative refinement.`
Date:
September 2012

Definition at line 183 of file cptrfs.f.

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 subroutine cpttrf ( integer N, real, dimension( * ) D, complex, dimension( * ) E, integer INFO )

CPTTRF

Purpose:
``` 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.```
Parameters:
 [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] 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.``` [in,out] 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.``` [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.```
Date:
September 2012

Definition at line 93 of file cpttrf.f.

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 subroutine cpttrs ( character UPLO, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB, integer INFO )

CPTTRS

Purpose:
``` 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.```
Parameters:
 [in] 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``` [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 factorization A = U**H*D*U or A = L*D*L**H.``` [in] 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.``` [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```
Date:
September 2012

Definition at line 122 of file cpttrs.f.

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 subroutine cptts2 ( integer IUPLO, integer N, integer NRHS, real, dimension( * ) D, complex, dimension( * ) E, complex, dimension( ldb, * ) B, integer LDB )

CPTTS2 solves a tridiagonal system of the form AX=B using the L D LH factorization computed by spttrf.

Purpose:
``` 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.```
Parameters:
 [in] 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``` [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 factorization A = U**H *D*U or A = L*D*L**H.``` [in] 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.``` [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).```
Date:
September 2012

Definition at line 114 of file cptts2.f.

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