.TH "pteqr" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME pteqr \- pteqr: eig, positive definite tridiagonal .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcpteqr\fP (compz, n, d, e, z, ldz, work, info)" .br .RI "\fBCPTEQR\fP " .ti -1c .RI "subroutine \fBdpteqr\fP (compz, n, d, e, z, ldz, work, info)" .br .RI "\fBDPTEQR\fP " .ti -1c .RI "subroutine \fBspteqr\fP (compz, n, d, e, z, ldz, work, info)" .br .RI "\fBSPTEQR\fP " .ti -1c .RI "subroutine \fBzpteqr\fP (compz, n, d, e, z, ldz, work, info)" .br .RI "\fBZPTEQR\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine cpteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, complex, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)" .PP \fBCPTEQR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> 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\&.) !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fICOMPZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the matrix\&. N >= 0\&. !> .fi .PP .br \fID\fP .PP .nf !> 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\&. !> .fi .PP .br \fIE\fP .PP .nf !> E is REAL array, dimension (N-1) !> On entry, the (n-1) subdiagonal elements of the tridiagonal !> matrix\&. !> On exit, E has been destroyed\&. !> .fi .PP .br \fIZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fILDZ\fP .PP .nf !> LDZ is INTEGER !> The leading dimension of the array Z\&. LDZ >= 1, and if !> COMPZ = 'V' or 'I', LDZ >= max(1,N)\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is REAL array, dimension (4*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> 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 leading principal !> minor of order i was not positive\&. !> > N the SVD algorithm failed to converge; !> if INFO = N+i, i off-diagonal elements of the !> bidiagonal factor did not converge to zero\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB144\fP of file \fBcpteqr\&.f\fP\&. .SS "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)" .PP \fBDPTEQR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> 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\&.) !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fICOMPZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the matrix\&. N >= 0\&. !> .fi .PP .br \fID\fP .PP .nf !> 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\&. !> .fi .PP .br \fIE\fP .PP .nf !> 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\&. !> .fi .PP .br \fIZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fILDZ\fP .PP .nf !> LDZ is INTEGER !> The leading dimension of the array Z\&. LDZ >= 1, and if !> COMPZ = 'V' or 'I', LDZ >= max(1,N)\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is DOUBLE PRECISION array, dimension (4*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> 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 leading principal !> minor of order i was not positive\&. !> > N the SVD algorithm failed to converge; !> if INFO = N+i, i off-diagonal elements of the !> bidiagonal factor did not converge to zero\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB144\fP of file \fBdpteqr\&.f\fP\&. .SS "subroutine spteqr (character compz, integer n, real, dimension( * ) d, real, dimension( * ) e, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)" .PP \fBSPTEQR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> SPTEQR computes all eigenvalues and, optionally, eigenvectors of a !> symmetric positive definite tridiagonal matrix by first factoring the !> matrix using SPTTRF, and then calling SBDSQR 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 SSYTRD, SSPTRD, or SSBTRD 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\&.) !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fICOMPZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the matrix\&. N >= 0\&. !> .fi .PP .br \fID\fP .PP .nf !> 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\&. !> .fi .PP .br \fIE\fP .PP .nf !> E is REAL array, dimension (N-1) !> On entry, the (n-1) subdiagonal elements of the tridiagonal !> matrix\&. !> On exit, E has been destroyed\&. !> .fi .PP .br \fIZ\fP .PP .nf !> Z is REAL 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\&. !> .fi .PP .br \fILDZ\fP .PP .nf !> LDZ is INTEGER !> The leading dimension of the array Z\&. LDZ >= 1, and if !> COMPZ = 'V' or 'I', LDZ >= max(1,N)\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is REAL array, dimension (4*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> 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 leading principal !> minor of order i was not positive\&. !> > N the SVD algorithm failed to converge; !> if INFO = N+i, i off-diagonal elements of the !> bidiagonal factor did not converge to zero\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB144\fP of file \fBspteqr\&.f\fP\&. .SS "subroutine zpteqr (character compz, integer n, double precision, dimension( * ) d, double precision, dimension( * ) e, complex*16, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)" .PP \fBZPTEQR\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a !> symmetric positive definite tridiagonal matrix by first factoring the !> matrix using DPTTRF and then calling ZBDSQR 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 ZHETRD, ZHPTRD, or ZHBTRD 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\&.) !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fICOMPZ\fP .PP .nf !> 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\&. !> .fi .PP .br \fIN\fP .PP .nf !> N is INTEGER !> The order of the matrix\&. N >= 0\&. !> .fi .PP .br \fID\fP .PP .nf !> 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\&. !> .fi .PP .br \fIE\fP .PP .nf !> 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\&. !> .fi .PP .br \fIZ\fP .PP .nf !> Z is COMPLEX*16 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\&. !> .fi .PP .br \fILDZ\fP .PP .nf !> LDZ is INTEGER !> The leading dimension of the array Z\&. LDZ >= 1, and if !> COMPZ = 'V' or 'I', LDZ >= max(1,N)\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is DOUBLE PRECISION array, dimension (4*N) !> .fi .PP .br \fIINFO\fP .PP .nf !> 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 leading principal !> minor of order i was not positive\&. !> > N the SVD algorithm failed to converge; !> if INFO = N+i, i off-diagonal elements of the !> bidiagonal factor did not converge to zero\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .PP Definition at line \fB144\fP of file \fBzpteqr\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.