Provided by: liblapack-doc_3.8.0-2_all

**NAME**

complexPTcomputational

**SYNOPSIS**

Functionssubroutinecptcon(N, D, E, ANORM, RCOND, RWORK, INFO)CPTCONsubroutinecpteqr(COMPZ, N, D, E, Z, LDZ, WORK, INFO)CPTEQRsubroutinecptrfs(UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX, FERR, BERR, WORK, RWORK, INFO)CPTRFSsubroutinecpttrf(N, D, E, INFO)CPTTRFsubroutinecpttrs(UPLO, N, NRHS, D, E, B, LDB, INFO)CPTTRSsubroutinecptts2(IUPLO, N, NRHS, D, E, B, LDB)CPTTS2solves 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** **Documentation**

subroutinecptcon(integerN,real,dimension(*)D,complex,dimension(*)E,realANORM,realRCOND,real,dimension(*)RWORK,integerINFO)CPTCONPurpose: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:NN is INTEGER The order of the matrix A. N >= 0.DD is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization of A, as computed by CPTTRF.EE 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.ANORMANORM is REAL The 1-norm of the original matrix A.RCONDRCOND 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.RWORKRWORK is REAL array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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.subroutinecpteqr(characterCOMPZ,integerN,real,dimension(*)D,real,dimension(*)E,complex,dimension(ldz,*)Z,integerLDZ,real,dimension(*)WORK,integerINFO)CPTEQRPurpose: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:COMPZCOMPZ 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.NN is INTEGER The order of the matrix. N >= 0.DD 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.EE is REAL array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix. On exit, E has been destroyed.ZZ 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if COMPZ = 'V' or 'I', LDZ >= max(1,N).WORKWORK is REAL array, dimension (4*N)INFOINFO 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:December 2016subroutinecptrfs(characterUPLO,integerN,integerNRHS,real,dimension(*)D,complex,dimension(*)E,real,dimension(*)DF,complex,dimension(*)EF,complex,dimension(ldb,*)B,integerLDB,complex,dimension(ldx,*)X,integerLDX,real,dimension(*)FERR,real,dimension(*)BERR,complex,dimension(*)WORK,real,dimension(*)RWORK,integerINFO)CPTRFSPurpose: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:UPLOUPLO 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.)NN is INTEGER The order of the matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.DD is REAL array, dimension (N) The n real diagonal elements of the tridiagonal matrix A.EE is COMPLEX array, dimension (N-1) The (n-1) off-diagonal elements of the tridiagonal matrix A (see UPLO).DFDF is REAL array, dimension (N) The n diagonal elements of the diagonal matrix D from the factorization computed by CPTTRF.EFEF 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).BB is COMPLEX array, dimension (LDB,NRHS) The right hand side matrix B.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).XX is COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CPTTRS. On exit, the improved solution matrix X.LDXLDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).FERRFERR 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).BERRBERR 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).WORKWORK is COMPLEX array, dimension (N)RWORKRWORK is REAL array, dimension (N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal valueInternalParameters: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:December 2016subroutinecpttrf(integerN,real,dimension(*)D,complex,dimension(*)E,integerINFO)CPTTRFPurpose: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:NN is INTEGER The order of the matrix A. N >= 0.DD 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.EE 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.INFOINFO 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:December 2016subroutinecpttrs(characterUPLO,integerN,integerNRHS,real,dimension(*)D,complex,dimension(*)E,complex,dimension(ldb,*)B,integerLDB,integerINFO)CPTTRSPurpose: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:UPLOUPLO 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 LNN is INTEGER The order of the tridiagonal matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.DD 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.EE 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.BB 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.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -k, the k-th argument had an illegal valueAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinecptts2(integerIUPLO,integerN,integerNRHS,real,dimension(*)D,complex,dimension(*)E,complex,dimension(ldb,*)B,integerLDB)CPTTS2solves 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:IUPLOIUPLO 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 LNN is INTEGER The order of the tridiagonal matrix A. N >= 0.NRHSNRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrix B. NRHS >= 0.DD 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.EE 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.BB 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.LDBLDB 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:June 2016

**Author**

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