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## man page of LAPACK-3

### LAPACK-3: computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A

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## NAME

LAPACK-3 - computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A## SYNOPSIS

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 ) CHARACTER JOBZ, RANGE, UPLO INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LRWORK, LWORK, M, N DOUBLE PRECISION ABSTOL, VL, VU INTEGER ISUPPZ( * ), IWORK( * ) DOUBLE PRECISION RWORK( * ), W( * ) COMPLEX*16 A( LDA, * ), WORK( * ), Z( LDZ, * )PURPOSE

ZHEEVR 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. 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 "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 : 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.## ARGUMENTS

JOBZ (input) CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. RANGE (input) 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. UPLO (input) CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored. N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) 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 (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). VL (input) DOUBLE PRECISION VU (input) DOUBLE PRECISION If RANGE='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'. IL (input) INTEGER IU (input) INTEGER If RANGE='I', the indices (in ascending order) of the smallest and largest eigenvalues 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 (input) 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 furutre 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 (output) INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. W (output) DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order. Z (output) 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 (input) INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N). ISUPPZ (output) 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 ). WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The length of the array WORK. LWORK >= max(1,2*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 (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) On exit, if INFO = 0, RWORK(1) returns the optimal (and minimal) LRWORK. 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 (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK. 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 (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error## FURTHER DETAILS

Based on contributions by 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 LAPACK driver routine (version 3.March 2011 LAPACK-3(3)

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