# heevr_2stage - Man Page

{he,sy}evr_2stage: eig, MRRR

## Synopsis

### Functions

subroutine **cheevr_2stage** (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)

**CHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices**

subroutine **dsyevr_2stage** (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

**DSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices**

subroutine **ssyevr_2stage** (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, iwork, liwork, info)

**SSYEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices**

subroutine **zheevr_2stage** (jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, isuppz, work, lwork, rwork, lrwork, iwork, liwork, info)

**ZHEEVR_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices**

## Detailed Description

## Function Documentation

### subroutine cheevr_2stage (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_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices**

**Purpose:**

CHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. CHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to CHETRD. Then, whenever possible, CHEEVR_2STAGE calls CSTEMR to compute eigenspectrum using Relatively Robust Representations. CSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' 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 CSTEMR's documentation and: - Inderjit S. Dhillon and Beresford N. Parlett: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : CHEEVR_2STAGE calls CSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. CHEEVR_2STAGE calls SSTEBZ and CSTEIN on non-ieee machines and when partial spectrum requests are made. 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.

**Parameters***JOBZ*JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.

*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

*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.

*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 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' 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, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', 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 REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.

*Z*Z is 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.

*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 dimension of the array WORK. If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 26*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 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. LRWORK >= max(1,24*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. LIWORK >= max(1,10*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 \n Osni Marques, LBNL/NERSC, USA \n Ken Stanley, Computer Science Division, University of California at Berkeley, USA \n Jason Riedy, Computer Science Division, University of California at Berkeley, USA \n

**Further Details:**

All details about the 2stage techniques are available in: Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line **402** of file **cheevr_2stage.f**.

### subroutine dsyevr_2stage (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_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices**

**Purpose:**

DSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. DSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to DSYTRD. Then, whenever possible, DSYEVR_2STAGE calls DSTEMR to compute the eigenspectrum using Relatively Robust Representations. DSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' 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: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : DSYEVR_2STAGE calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSYEVR_2STAGE calls DSTEBZ and SSTEIN 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. Not available in this release.

*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

*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 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| ) , 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 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' 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, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', 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 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 JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 26*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + 5*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + 5*N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 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 LWORK.

*LIWORK*LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*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 = 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 \n Osni Marques, LBNL/NERSC, USA \n Ken Stanley, Computer Science Division, University of California at Berkeley, USA \n Jason Riedy, Computer Science Division, University of California at Berkeley, USA \n

**Further Details:**

All details about the 2stage techniques are available in: Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line **378** of file **dsyevr_2stage.f**.

### subroutine ssyevr_2stage (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_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices**

**Purpose:**

SSYEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. SSYEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to SSYTRD. Then, whenever possible, SSYEVR_2STAGE calls SSTEMR to compute the eigenspectrum using Relatively Robust Representations. SSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' 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 SSTEMR's documentation and: - Inderjit S. Dhillon and Beresford N. Parlett: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : SSYEVR_2STAGE calls SSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. SSYEVR_2STAGE 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.

**Parameters***JOBZ*JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.

*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 SSTEIN are called

*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 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.

*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 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' 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, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', 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 REAL array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.

*Z*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*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 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*WORK is REAL 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 JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 26*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + 5*N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + 5*N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 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 LWORK.

*LIWORK*LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*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 = 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 \n Osni Marques, LBNL/NERSC, USA \n Ken Stanley, Computer Science Division, University of California at Berkeley, USA \n Jason Riedy, Computer Science Division, University of California at Berkeley, USA \n

**Further Details:**

All details about the 2stage techniques are available in: Azzam Haidar, Hatem Ltaief, and Jack Dongarra. Parallel reduction to condensed forms for symmetric eigenvalue problems using aggregated fine-grained and memory-aware kernels. In Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '11), New York, NY, USA, Article 8 , 11 pages. http://doi.acm.org/10.1145/2063384.2063394 A. Haidar, J. Kurzak, P. Luszczek, 2013. An improved parallel singular value algorithm and its implementation for multicore hardware, In Proceedings of 2013 International Conference for High Performance Computing, Networking, Storage and Analysis (SC '13). Denver, Colorado, USA, 2013. Article 90, 12 pages. http://doi.acm.org/10.1145/2503210.2503292 A. Haidar, R. Solca, S. Tomov, T. Schulthess and J. Dongarra. A novel hybrid CPU-GPU generalized eigensolver for electronic structure calculations based on fine-grained memory aware tasks. International Journal of High Performance Computing Applications. Volume 28 Issue 2, Pages 196-209, May 2014. http://hpc.sagepub.com/content/28/2/196

Definition at line **378** of file **ssyevr_2stage.f**.

### subroutine zheevr_2stage (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_2STAGE computes the eigenvalues and, optionally, the left and/or right eigenvectors for HE matrices**

**Purpose:**

ZHEEVR_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A using the 2stage technique for the reduction to tridiagonal. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. ZHEEVR_2STAGE first reduces the matrix A to tridiagonal form T with a call to ZHETRD. Then, whenever possible, ZHEEVR_2STAGE calls ZSTEMR to compute eigenspectrum using Relatively Robust Representations. ZSTEMR computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various 'good' 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: 'Multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices,' Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004. - Inderjit Dhillon and Beresford Parlett: 'Orthogonal Eigenvectors and Relative Gaps,' SIAM Journal on Matrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154. - Inderjit Dhillon: 'A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem', Computer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997. Note 1 : ZHEEVR_2STAGE calls ZSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. ZHEEVR_2STAGE 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**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

*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 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.

*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**IU**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| ) , 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 'Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,' 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, 'Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices', LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.

*M**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.

*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*LWORK is INTEGER The dimension of the array WORK. If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 26*N, dimension) where dimension = max(stage1,stage2) + (KD+1)*N + N = N*KD + N*max(KD+1,FACTOPTNB) + max(2*KD*KD, KD*NTHREADS) + (KD+1)*N + N where KD is the blocking size of the reduction, FACTOPTNB is the blocking used by the QR or LQ algorithm, usually FACTOPTNB=128 is a good choice NTHREADS is the number of threads used when openMP compilation is enabled, otherwise =1. If JOBZ = 'V' and N > 1, LWORK must be queried. Not yet available 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 The length of the array RWORK. LRWORK >= max(1,24*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. LIWORK >= max(1,10*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***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

**Further Details:**

Definition at line **402** of file **zheevr_2stage.f**.

## Author

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