heevr(3)

📦

version 3.12.1 · 9 versions

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🎀 heevr(3) 🎀

okay so bestie, this is heevr(3) from the Library Functions Manual — the eig-with-MRRR situation, and it's genuinely one of the prettier corners of LAPACK. It's the family page for {he,sy}evr and it's lowkey iconic: four routines, one vibe, and a whole Relatively Robust Representations flex. Sit down, hydrate, we're doing this 🫶

✨ NAME ✨

heevr — {he,sy}evr: eig, MRRR

🔥 SYNOPSIS 🔥

Functions

cheevr — CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

subroutine cheevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
    m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork,
    info)

dsyevr — DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

subroutine dsyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
    m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

ssyevr — SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

subroutine ssyevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
    m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

zheevr — ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

subroutine zheevr (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
    m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork,
    info)

💜 Detailed Description 💜

No cap, this page is mostly the docs for each of the four routines underneath the family umbrella. Scroll on, bestie.

🌸 Function Documentation 🌸

`cheevr`

subroutine cheevr (character jobz, character range, character uplo, integer
    n, complex, dimension( lda, * ) a, integer lda, real vl, real vu,
    integer il, integer iu, real abstol, integer m, real, dimension( * ) w,
    complex, dimension( ldz, * ) z, integer ldz, integer, dimension( * )
    isuppz, complex, dimension( * ) work, integer lwork, real, dimension( *
    ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork,
    integer info)

CHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices 🦋

Purpose

okay so the whole point 💅 — CHEEVR computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. And the selecting part is the main character energy here: you can pick your eigenvalues either by specifying a range of values or a range of indices. Two completely different knobs, both valid.

⚠️ heads up: invocations with different choices for these parameters may result in the computation of slightly different eigenvalues and/or eigenvectors for the same matrix. Not a bug, bestie — it's because there's a whole variety of algorithms (each one performing best for a particular set of options), and CHEEVR attempts to select the best one based on the various parameters. In all cases the computed values are accurate within the limits of finite precision arithmetic. periodt.

So how does it actually do the thing? CHEEVR first reduces the matrix A to tridiagonal form T with a call to CHETRD. Then, whenever possible, CHEEVR calls CSTEMR to compute the eigenspectrum using Relatively Robust Representations. CSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible — yes, the algorithm is dodging Gram-Schmidt, we stan efficiency 🌈

More specifically, the various steps of the algorithm are as follows 👇

For each unreduced block (submatrix) of T:

(a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL ✨

For more details, see CSTEMR's documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

💀 Note 1: CHEEVR calls CSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made.

🫶 Note 2, and this one matters: normal execution of CSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. Not a drill, know your environment.

Parameters

JOBZ — JOBZ is CHARACTER*1

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

This parameter influences the choice of the algorithm and may alter the computed values. (yes, like we said — the algorithm vibe check changes the numbers, slightly)

RANGE — RANGE is CHARACTER*1

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.

For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and CSTEIN are called.

This parameter influences the choice of the algorithm and may alter the computed values. (again — same deal, this one's a main character too)

UPLO — UPLO is CHARACTER*1

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N — N is INTEGER

The order of the matrix A. N >= 0.

A — A is COMPLEX array, dimension (LDA, N)

On entry, the Hermitian matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A.

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed 😭 yes, she takes, she does not give back

LDA — LDA is INTEGER

The leading dimension of the array A. LDA >= max(1,N).

VL — VL is REAL

If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

VU — VU is REAL

If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

IL — IL is INTEGER

If RANGE='I', the index of the smallest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

IU — IU is INTEGER

If RANGE='I', the index of the largest eigenvalue to be returned.

1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

ABSTOL — ABSTOL is REAL

The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

See by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

🧚 tiny note bestie: that "current code does not make any guarantees" line is doing a lot of heavy lifting — read it twice before you go promising anybody high relative accuracy today.

M — M is INTEGER

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W — W is REAL array, dimension (N)

The first M elements contain the selected eigenvalues in ascending order. (ascending!! she's tidy like that 💅)

Z

okay so Z is literally the next thing and this is exactly where part 1 cuts off — not me being mysterious, that's just the page. She's coming right up in part 2 🌸

so Z is a COMPLEX array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe. 🫶

`LDZ`

LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). ✨

`ISUPPZ`

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ). The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of CSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the unitary transformations applied by CUNMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1. 💅

`WORK`

WORK is COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 🔥

`LWORK`

LWORK is INTEGER. The length of the array WORK. If N <= 1, LWORK >= 1, else LWORK >= 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for CHETRD and for CUNMTR as returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 😭

`RWORK`

RWORK is REAL array, dimension (MAX(1,LRWORK)). On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK. 🌸

`LRWORK`

LRWORK is INTEGER. The length of the array RWORK. If N <= 1, LRWORK >= 1, else LRWORK >= 24*N. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 🦋

`IWORK`

IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK. 🎀

`LIWORK`

LIWORK is INTEGER. The dimension of the array IWORK. If N <= 1, LIWORK >= 1, else LIWORK >= 10*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA. 💖

`INFO`

INFO is INTEGER. = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: Internal error. 👑

Author

Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver, NAG Ltd. 🏫

Contributors:

Inderjit Dhillon, IBM Almaden, USA; Osni Marques, LBNL/NERSC, USA; Ken Stanley, Computer Science Division, University of California at Berkeley, USA; Jason Riedy, Computer Science Division, University of California at Berkeley, USA. 🫶

Definition at line 371 of file `cheevr.f`. ⭐

`dsyevr`

subroutine dsyevr (character jobz, character range, character uplo, integer
    n, double precision, dimension( lda, * ) a, integer lda, double
    precision vl, double precision vu, integer il, integer iu, double
    precision abstol, integer m, double precision, dimension( * ) w, double
    precision, dimension( ldz, * ) z, integer ldz, integer, dimension( * )
    isuppz, double precision, dimension( * ) work, integer lwork, integer,
    dimension( * ) iwork, integer liwork, integer info)

DSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices. 🌀

Purpose

DSYEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Invocations with different choices for these parameters may result in the computation of slightly different eigenvalues and/or eigenvectors for the same matrix. The reason for this behavior is that there exists a variety of algorithms (each performing best for a particular set of options) with DSYEVR attempting to select the best based on the various parameters. In all cases, the computed values are accurate within the limits of finite precision arithmetic. 🧚

DSYEVR first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR calls DSTEMR to compute the eigenspectrum using Relatively Robust Representations. DSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. 💫

For each unreduced block (submatrix) of T:

(a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general. 🦋
(b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d). 🌸
(c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy. 🎀
(d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain. 💖

The desired accuracy of the output can be specified by the input parameter ABSTOL. ✨

For more details, see DSTEMR's documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1: DSYEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSYEVR calls DSTEBZ and DSTEIN on non-ieee machines and when partial spectrum requests are made. ⚠️

Normal execution of DSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. 💀

Parameters

`JOBZ`

JOBZ is CHARACTER*1. = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. This parameter influences the choice of the algorithm and may alter the computed values. 👑

`RANGE`

RANGE is CHARACTER*1. = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found. For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and DSTEIN are called. This parameter influences the choice of the algorithm and may alter the computed values. 🌈

`UPLO`

UPLO is CHARACTER*1. = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. 💅

`N`

N is INTEGER. The order of the matrix A. N >= 0. 🎀

`A`

A is DOUBLE PRECISION array, dimension (LDA, N). On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 🔥

`LDA`

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,N). ✨

`VL`

VL — VL is DOUBLE PRECISION. bestie this is only relevant when RANGE='V', and it's the lower bound of the interval we're out here searching for eigenvalues in 🌸. Rule of the land: VL < VU. Not referenced at all if RANGE = 'A' or 'I' (so like, don't stress about it then 💅).
VU — VU is DOUBLE PRECISION. If RANGE='V', this is the upper bound of the interval being searched for eigenvalues. Same deal: VL < VU. Not referenced if RANGE = 'A' or 'I' 🦋.
IL — IL is INTEGER. If RANGE='I', this is the index of the smallest eigenvalue to be returned. Constraints fr fr: 1 <= IL <= IU <= N, if N > 0; and IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V' ✨.
IU — IU is INTEGER. If RANGE='I', this is the index of the largest eigenvalue to be returned. Same constraints: 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V' ✨.
ABSTOL — ABSTOL is DOUBLE PRECISION. It's the absolute error tolerance for the eigenvalues 👑. An approximate eigenvalue gets accepted as converged when it's determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

See by Demmel and Kahan, LAPACK Working Note #3. 📚

If high relative accuracy is important to you (and honestly, when is it not), set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will 🫶. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

M — M is INTEGER. The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', then M = N, and if RANGE = 'I', then M = IU-IL+1 💖.
W — W is DOUBLE PRECISION array, dimension (N). The first M elements contain the selected eigenvalues in ascending order ⭐.
Z — Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M)). If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i) 🦋. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe 💜.
LDZ — LDZ is INTEGER. The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N) 🎀.
ISUPPZ — ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ). The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of DSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by DORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1 🧚‍♀️.
WORK — WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK 🌈.
LWORK — LWORK is INTEGER. The dimension of the array WORK. If N <= 1, LWORK >= 1, else LWORK >= 26*N. For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for DSYTRD and DORMTR returned by ILAENV 🔥.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA 💅.

IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK 🌸.
LIWORK — LIWORK is INTEGER. The dimension of the array IWORK. If N <= 1, LIWORK >= 1, else LIWORK >= 10*N 🍓.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the IWORK array, returns this value as the first entry of the IWORK array, and no error message related to LIWORK is issued by XERBLA 💅.

INFO — INFO is INTEGER. The vibe check on how it went:
- = 0: successful exit 🎉
- < 0: if INFO = -i, the i-th argument had an illegal value 😭
- > 0: Internal error 💀

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 Contributors:

Inderjit Dhillon, IBM Almaden, USA

Osni Marques, LBNL/NERSC, USA

Ken Stanley, Computer Science Division, University of California at Berkeley, USA

Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 346 of file dsyevr.f.

💜 subroutine `ssyevr` — the single-precision sibling, and yes she ate

subroutine ssyevr (character jobz, character range, character uplo, integer
    n, real, dimension( lda, * ) a, integer lda, real vl, real vu, integer
    il, integer iu, real abstol, integer m, real, dimension( * ) w, real,
    dimension( ldz, * ) z, integer ldz, integer, dimension( * ) isuppz,
    real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork,
    integer liwork, integer info)

SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices ✨

🦋 Purpose:

SSYEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Quick heads-up bestie: invocations with different choices for these parameters may result in the computation of slightly different eigenvalues and/or eigenvectors for the same matrix 😭. The reason for this behavior is that there exists a variety of algorithms (each performing best for a particular set of options) with SSYEVR attempting to select the best based on the various parameters. In all cases, the computed values are accurate within the limits of finite precision arithmetic 🙈.

SSYEVR first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations 🧚. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows 👇

For each unreduced block (submatrix) of T:

Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general 👀.
Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy 🎀.
For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain 🔁.

The desired accuracy of the output can be specified by the input parameter ABSTOL ⭐.

For more details, see SSTEMR's documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1: SSYEVR calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and when partial spectrum requests are made 💀

Normal execution of SSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. Not me warning you twice, this is a real one 😭

🌸 Parameters

JOBZ

JOBZ is CHARACTER*1 — okay so this one is the eigenvector switch, bestie:

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

This parameter influences the choice of the algorithm and may alter the computed values.

RANGE (CHARACTER*1) — this is the “which eigenvalues are we even looking for” dial:

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.

For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and SSTEIN are called.

This parameter influences the choice of the algorithm and may alter the computed values.

UPLO (CHARACTER*1) — which triangle of A is the main character:

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

N is INTEGER — the order of the matrix A. N >= 0.

A is REAL array, dimension (LDA, N) — on entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. Gone. Bestie, it’s gone.

LDA is INTEGER — the leading dimension of the array A. LDA >= max(1,N).

VL is REAL — if RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.

VU is REAL — if RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.

IL is INTEGER — if RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

IU is INTEGER — if RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.

ABSTOL is REAL — the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

The docs also say: See by Demmel and Kahan, LAPACK Working Note #3.

M is INTEGER — the total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

W is REAL array, dimension (N) — the first M elements contain the selected eigenvalues in ascending order.

Z is REAL array, dimension (LDZ, max(1,M)) — if JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe.

LDZ is INTEGER — the leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) — the support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by SORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1.

WORK is REAL array, dimension (MAX(1,LWORK)) — on exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK is INTEGER — the dimension of the array WORK. If N <= 1, LWORK >= 1, else LWORK >= 26*N. For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for SSYTRD and SORMTR returned by ILAENV.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

IWORK is INTEGER array, dimension (MAX(1,LIWORK)) — on exit, if INFO = 0, IWORK(1) returns the optimal LWORK.

LIWORK is INTEGER — the dimension of the array IWORK. If N <= 1, LIWORK >= 1, else LIWORK >= 10*N.

If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.

INFO is INTEGER:

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: Internal error

Author ✨

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Contributors: 🫶

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 348 of file ssyevr.f.

subroutine `zheevr` 💅

   subroutine zheevr (character jobz, character range, character uplo, integer
       n, complex*16, dimension( lda, * ) a, integer lda, double precision vl,
       double precision vu, integer il, integer iu, double precision abstol,
       integer m, double precision, dimension( * ) w, complex*16, dimension(
       ldz, * ) z, integer ldz, integer, dimension( * ) isuppz, complex*16,
       dimension( * ) work, integer lwork, double precision, dimension( * )
       rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork,
       integer info)

ZHEEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices

Purpose

okay so bestie, here’s the whole ZHEEVR situation: it computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Invocations with different choices for these parameters may result in the computation of slightly different eigenvalues and/or eigenvectors for the same matrix. Why? Because there’s a whole variety of algorithms (each performing best for a particular set of options) with ZHEEVR attempting to select the best based on the various parameters. In all cases, the computed values are accurate within the limits of finite precision arithmetic. periodt. 💅

ZHEEVR first reduces the matrix A to tridiagonal form T with a call to ZHETRD. Then, whenever possible, ZHEEVR calls ZSTEMR to compute eigenspectrum using Relatively Robust Representations. ZSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows.

For each unreduced block (submatrix) of T,

(a) Compute T - sigma I = L D L^T, so that L and D define all the wanted eigenvalues to high relative accuracy. This means that small relative changes in the entries of D and L cause only small relative changes in the eigenvalues and eigenvectors. The standard (unfactored) representation of the tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full accuracy of the computed eigenvalues only right before the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization, and refine the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute the corresponding eigenvector by forming a rank revealing twisted factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input parameter ABSTOL.

For more details, see ZSTEMR’s documentation and:

Inderjit S. Dhillon and Beresford N. Parlett: Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
Inderjit Dhillon and Beresford Parlett: SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.
Inderjit Dhillon: , Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.

Note 1 : ZHEEVR calls ZSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR calls DSTEBZ and ZSTEIN on non-ieee machines and when partial spectrum requests are made.

Normal execution of ZSTEMR may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner. 😭💀

💅 Parameters 💅

💖 `JOBZ` 💖

JOBZ is CHARACTER*1.

= 'N': Compute eigenvalues only;
= 'V': Compute eigenvalues and eigenvectors.

This parameter influences the choice of the algorithm and may alter the computed values.

🌸 `RANGE` 🌸

RANGE is CHARACTER*1.

= 'A': all eigenvalues will be found.
= 'V': all eigenvalues in the half-open interval (VL,VU] will be found.
= 'I': the IL-th through IU-th eigenvalues will be found.

For RANGE = 'V' or 'I' and IU - IL < N - 1, DSTEBZ and ZSTEIN are called.

This parameter influences the choice of the algorithm and may alter the computed values.

🦋 `UPLO` 🦋

UPLO is CHARACTER*1.

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.

🎀 `N` 🎀

N is INTEGER.

The order of the matrix A. N >= 0.

💜 `A` 💜

A is COMPLEX*16 array, dimension (LDA, N).

On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed. 😭

⭐ `LDA` ⭐

LDA is INTEGER.

The leading dimension of the array A. LDA >= max(1,N).

🍓 `VL` 🍓

VL is DOUBLE PRECISION.

If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

🍓 `VU` 🍓

VU is DOUBLE PRECISION.

If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU.

Not referenced if RANGE = 'A' or 'I'.

✨ `IL` ✨

IL is INTEGER.

If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

✨ `IU` ✨

IU is INTEGER.

If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.

Not referenced if RANGE = 'A' or 'V'.

🌈 `ABSTOL` 🌈

ABSTOL is DOUBLE PRECISION.

The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

ABSTOL + EPS *   max( |a|,|b| ) ,

See by Demmel and Kahan, LAPACK Working Note #3.

If high relative accuracy is important, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, , LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

👑 `M` 👑

M is INTEGER.

The total number of eigenvalues found. 0 <= M <= N.

If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

🧚 `W` 🧚

W is DOUBLE PRECISION array, dimension (N).

The first M elements contain the selected eigenvalues in ascending order.

🫶 `Z` 🫶

Z is COMPLEX*16 array, dimension (LDZ, max(1,M)).

If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i).

If JOBZ = 'N', then Z is not referenced.

Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used. Supplying N columns is always safe. 💅

🎀 `LDZ` 🎀

LDZ is INTEGER.

The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).

🌟 `ISUPPZ` 🌟

ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ).

The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of ZSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the unitary transformations applied by ZUNMTR.

Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1.

💫 `WORK` 💫

WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)).

On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

💫 `LWORK` 💫

okay bestie, we're still in the parameter list — almost there! 🫶

LWORK — LWORK is INTEGER. It's the length of the array WORK. If N <= 1, then LWORK >= 1; otherwise LWORK >= 2*N. For optimal efficiency, LWORK >= (NB+1)*N, where NB is the max of the blocksize for ZHETRD and for ZUNMTR as returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
RWORK — RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)). On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK.
LRWORK — LRWORK is INTEGER. It's the length of the array RWORK. If N <= 1, then LRWORK >= 1; otherwise LRWORK >= 24*N. If LRWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
IWORK — IWORK is INTEGER array, dimension (MAX(1,LIWORK)). On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK.
LIWORK — LIWORK is INTEGER. It's the dimension of the array IWORK. If N <= 1, then LIWORK >= 1; otherwise LIWORK >= 10*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK, RWORK and IWORK arrays, returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK or LRWORK or LIWORK is issued by XERBLA.
INFO — INFO is INTEGER. It's the exit vibe check:
- = 0: successful exit — slay ✨
- < 0: if INFO = -i, the i-th argument had an illegal value — oops 🙈
- > 0: Internal error — not iconic 💀

Author 💖

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Contributors 🫶

Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 371 of file zheevr.f.

Author (the auto-generated one) 🤖

Generated automatically by Doxygen for LAPACK from the source code.

That's the whole page, bestie! 🎀 From LAPACK Version 3.12.0, the heevr(3) page — date not shown in the footer, so we're calling it timeless 💅✨

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

heevr(3)

🎀 heevr(3) 🎀

✨ NAME ✨

🔥 SYNOPSIS 🔥

Functions

💜 Detailed Description 💜

🌸 Function Documentation 🌸

cheevr

Purpose

Parameters

LDZ

ISUPPZ

WORK

LWORK

RWORK

LRWORK

IWORK

LIWORK

INFO

Author

Contributors:

Definition at line 371 of file cheevr.f. ⭐

dsyevr

Purpose

Parameters

JOBZ

RANGE

UPLO

N

A

LDA

VL

👑 Author

🌈 Contributors:

💜 subroutine ssyevr — the single-precision sibling, and yes she ate

🦋 Purpose:

🌸 Parameters

Author ✨

Contributors: 🫶

subroutine zheevr 💅

Purpose

💅 Parameters 💅

💖 JOBZ 💖

🌸 RANGE 🌸

🦋 UPLO 🦋

🎀 N 🎀

💜 A 💜

⭐ LDA ⭐

🍓 VL 🍓

🍓 VU 🍓

✨ IL ✨

✨ IU ✨

🌈 ABSTOL 🌈

👑 M 👑

🧚 W 🧚

🫶 Z 🫶

🎀 LDZ 🎀

🌟 ISUPPZ 🌟

💫 WORK 💫

💫 LWORK 💫

Author 💖

Contributors 🫶

Author (the auto-generated one) 🤖

`cheevr`

`LDZ`

`ISUPPZ`

`WORK`

`LWORK`

`RWORK`

`LRWORK`

`IWORK`

`LIWORK`

`INFO`

Definition at line 371 of file `cheevr.f`. ⭐

`dsyevr`

`JOBZ`

`RANGE`

`UPLO`

`N`

`A`

`LDA`

`VL`

💜 subroutine `ssyevr` — the single-precision sibling, and yes she ate

subroutine `zheevr` 💅

💖 `JOBZ` 💖

🌸 `RANGE` 🌸

🦋 `UPLO` 🦋

🎀 `N` 🎀

💜 `A` 💜

⭐ `LDA` ⭐

🍓 `VL` 🍓

🍓 `VU` 🍓

✨ `IL` ✨

✨ `IU` ✨

🌈 `ABSTOL` 🌈

👑 `M` 👑

🧚 `W` 🧚

🫶 `Z` 🫶

🎀 `LDZ` 🎀

🌟 `ISUPPZ` 🌟

💫 `WORK` 💫

💫 `LWORK` 💫