--- name: "pteqr" section: "3" platform: "linux" package: "lapack-doc" version: "3.12.1-2" release: "Arch Linux 2026-11-01" platforms: ["linux"] url: "https://new.carta.tech/pteqr" --- # pteqr(3) pteqr: eig, positive definite tridiagonal ## NAME pteqr - pteqr: eig, positive definite tridiagonal ## SYNOPSIS ### Functions subroutine **cpteqr** (compz, n, d, e, z, ldz, work, info)\ **CPTEQR** subroutine **dpteqr** (compz, n, d, e, z, ldz, work, info)\ **DPTEQR** subroutine **spteqr** (compz, n, d, e, z, ldz, work, info)\ **SPTEQR** subroutine **zpteqr** (compz, n, d, e, z, ldz, work, info)\ **ZPTEQR** ## Detailed Description ## Function Documentation ### 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** *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 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **144** of file **cpteqr.f**. ### 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** **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** *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. !> ``` *N* ``` !> N is INTEGER !> The order of the matrix. N >= 0. !> ``` *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. !> ``` *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. !> ``` *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. !> ``` *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 DOUBLE PRECISION 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 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **144** of file **dpteqr.f**. ### subroutine spteqr (character compz, integer n, real, dimension( \* ) d, real, dimension( \* ) e, real, dimension( ldz, \* ) z, integer ldz, real, dimension( \* ) work, integer info) **SPTEQR** **Purpose:** ``` !> !> 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.) !> ``` **Parameters** *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. !> ``` *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 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. !> ``` *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 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **144** of file **spteqr.f**. ### 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) **ZPTEQR** **Purpose:** ``` !> !> 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.) !> ``` **Parameters** *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 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. !> ``` *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. !> ``` *Z* ``` !> 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. !> ``` *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 DOUBLE PRECISION 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 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. !> ``` **Author** Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Definition at line **144** of file **zpteqr.f**. ## Author Generated automatically by Doxygen for LAPACK from the source code.