USAGE: work, rwork, iwork, info, d, e, z = NumRu::Lapack.cstedc( compz, d, e, z, [:lwork => lwork, :lrwork => lrwork, :liwork => liwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, INFO ) * Purpose * ======= * * CSTEDC computes all eigenvalues and, optionally, eigenvectors of a * symmetric tridiagonal matrix using the divide and conquer method. * The eigenvectors of a full or band complex Hermitian matrix can also * be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this * matrix to tridiagonal form. * * This code makes very mild assumptions about floating point * arithmetic. It will work on machines with a guard digit in * add/subtract, or on those binary machines without guard digits * which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. * It could conceivably fail on hexadecimal or decimal machines * without guard digits, but we know of none. See SLAED3 for details. * * Arguments * ========= * * COMPZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only. * = 'I': Compute eigenvectors of tridiagonal matrix also. * = 'V': Compute eigenvectors of original Hermitian matrix * also. On entry, Z contains the unitary matrix used * to reduce the original matrix to tridiagonal form. * * N (input) INTEGER * The dimension of the symmetric tridiagonal matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the diagonal elements of the tridiagonal matrix. * On exit, if INFO = 0, the eigenvalues in ascending order. * * E (input/output) REAL array, dimension (N-1) * On entry, the subdiagonal elements of the tridiagonal matrix. * On exit, E has been destroyed. * * Z (input/output) COMPLEX array, dimension (LDZ,N) * On entry, if COMPZ = 'V', then Z contains the unitary * matrix used in the reduction to tridiagonal form. * On exit, if INFO = 0, then if COMPZ = 'V', Z contains the * orthonormal eigenvectors of the original Hermitian matrix, * and if COMPZ = 'I', Z contains the orthonormal eigenvectors * of the symmetric tridiagonal matrix. * If COMPZ = 'N', then Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1. * If eigenvectors are desired, then LDZ >= max(1,N). * * WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. * If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1. * If COMPZ = 'V' and N > 1, LWORK must be at least N*N. * Note that for COMPZ = 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LWORK need * only be 1. * * 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) REAL array, dimension (MAX(1,LRWORK)) * On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. * * LRWORK (input) INTEGER * The dimension of the array RWORK. * If COMPZ = 'N' or N <= 1, LRWORK must be at least 1. * If COMPZ = 'V' and N > 1, LRWORK must be at least * 1 + 3*N + 2*N*lg N + 3*N**2 , * where lg( N ) = smallest integer k such * that 2**k >= N. * If COMPZ = 'I' and N > 1, LRWORK must be at least * 1 + 4*N + 2*N**2 . * Note that for COMPZ = 'I' or 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LRWORK * need only be max(1,2*(N-1)). * * 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 LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. * If COMPZ = 'N' or N <= 1, LIWORK must be at least 1. * If COMPZ = 'V' or N > 1, LIWORK must be at least * 6 + 6*N + 5*N*lg N. * If COMPZ = 'I' or N > 1, LIWORK must be at least * 3 + 5*N . * Note that for COMPZ = 'I' or 'V', then if N is less than or * equal to the minimum divide size, usually 25, then LIWORK * need only be 1. * * 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: The algorithm failed to compute an eigenvalue while * working on the submatrix lying in rows and columns * INFO/(N+1) through mod(INFO,N+1). * * Further Details * =============== * * Based on contributions by * Jeff Rutter, Computer Science Division, University of California * at Berkeley, USA * * ===================================================================== *go to the page top

USAGE: m, w, z, isuppz, work, iwork, info, d, e = NumRu::Lapack.cstegr( jobz, range, d, e, vl, vu, il, iu, abstol, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CSTEGR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO ) * Purpose * ======= * * CSTEGR computes selected eigenvalues and, optionally, eigenvectors * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has * a well defined set of pairwise different real eigenvalues, the corresponding * real eigenvectors are pairwise orthogonal. * * The spectrum may be computed either completely or partially by specifying * either an interval (VL,VU] or a range of indices IL:IU for the desired * eigenvalues. * * CSTEGR is a compatability wrapper around the improved CSTEMR routine. * See SSTEMR for further details. * * One important change is that the ABSTOL parameter no longer provides any * benefit and hence is no longer used. * * Note : CSTEGR and CSTEMR work only on machines which follow * IEEE-754 floating-point standard in their handling of infinities and * NaNs. Normal execution may create these exceptiona values and hence * may abort due to a floating point exception in environments which * do not conform to the IEEE-754 standard. * * 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. * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the N diagonal elements of the tridiagonal matrix * T. On exit, D is overwritten. * * E (input/output) REAL array, dimension (N) * On entry, the (N-1) subdiagonal elements of the tridiagonal * matrix T in elements 1 to N-1 of E. E(N) need not be set on * input, but is used internally as workspace. * On exit, E is overwritten. * * VL (input) REAL * VU (input) REAL * 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. * Not referenced if RANGE = 'A' or 'V'. * * ABSTOL (input) REAL * Unused. Was the absolute error tolerance for the * eigenvalues/eigenvectors in previous versions. * * 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) REAL array, dimension (N) * The first M elements contain the selected eigenvalues in * ascending order. * * Z (output) COMPLEX array, dimension (LDZ, max(1,M) ) * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z * contain the orthonormal eigenvectors of the matrix T * 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 (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', then 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 computed eigenvector * is nonzero only in elements ISUPPZ( 2*i-1 ) through * ISUPPZ( 2*i ). This is relevant in the case when the matrix * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal * (and minimal) LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,18*N) * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. * 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 (workspace/output) INTEGER array, dimension (LIWORK) * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. LIWORK >= max(1,10*N) * if the eigenvectors are desired, and LIWORK >= max(1,8*N) * if only the eigenvalues are to be computed. * 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 (output) INTEGER * On exit, INFO * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = 1X, internal error in SLARRE, * if INFO = 2X, internal error in CLARRV. * Here, the digit X = ABS( IINFO ) < 10, where IINFO is * the nonzero error code returned by SLARRE or * CLARRV, respectively. * * Further Details * =============== * * Based on contributions by * Inderjit Dhillon, IBM Almaden, USA * Osni Marques, LBNL/NERSC, USA * Christof Voemel, LBNL/NERSC, USA * * ===================================================================== * * .. Local Scalars .. LOGICAL TRYRAC * .. * .. External Subroutines .. EXTERNAL CSTEMR * ..go to the page top

USAGE: z, ifail, info = NumRu::Lapack.cstein( d, e, w, iblock, isplit, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO ) * Purpose * ======= * * CSTEIN computes the eigenvectors of a real symmetric tridiagonal * matrix T corresponding to specified eigenvalues, using inverse * iteration. * * The maximum number of iterations allowed for each eigenvector is * specified by an internal parameter MAXITS (currently set to 5). * * Although the eigenvectors are real, they are stored in a complex * array, which may be passed to CUNMTR or CUPMTR for back * transformation to the eigenvectors of a complex Hermitian matrix * which was reduced to tridiagonal form. * * * Arguments * ========= * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the tridiagonal matrix T. * * E (input) REAL array, dimension (N-1) * The (n-1) subdiagonal elements of the tridiagonal matrix * T, stored in elements 1 to N-1. * * M (input) INTEGER * The number of eigenvectors to be found. 0 <= M <= N. * * W (input) REAL array, dimension (N) * The first M elements of W contain the eigenvalues for * which eigenvectors are to be computed. The eigenvalues * should be grouped by split-off block and ordered from * smallest to largest within the block. ( The output array * W from SSTEBZ with ORDER = 'B' is expected here. ) * * IBLOCK (input) INTEGER array, dimension (N) * The submatrix indices associated with the corresponding * eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to * the first submatrix from the top, =2 if W(i) belongs to * the second submatrix, etc. ( The output array IBLOCK * from SSTEBZ is expected here. ) * * ISPLIT (input) INTEGER array, dimension (N) * The splitting points, at which T breaks up into submatrices. * The first submatrix consists of rows/columns 1 to * ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1 * through ISPLIT( 2 ), etc. * ( The output array ISPLIT from SSTEBZ is expected here. ) * * Z (output) COMPLEX array, dimension (LDZ, M) * The computed eigenvectors. The eigenvector associated * with the eigenvalue W(i) is stored in the i-th column of * Z. Any vector which fails to converge is set to its current * iterate after MAXITS iterations. * The imaginary parts of the eigenvectors are set to zero. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= max(1,N). * * WORK (workspace) REAL array, dimension (5*N) * * IWORK (workspace) INTEGER array, dimension (N) * * IFAIL (output) INTEGER array, dimension (M) * On normal exit, all elements of IFAIL are zero. * If one or more eigenvectors fail to converge after * MAXITS iterations, then their indices are stored in * array IFAIL. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, then i eigenvectors failed to converge * in MAXITS iterations. Their indices are stored in * array IFAIL. * * Internal Parameters * =================== * * MAXITS INTEGER, default = 5 * The maximum number of iterations performed. * * EXTRA INTEGER, default = 2 * The number of iterations performed after norm growth * criterion is satisfied, should be at least 1. * * ===================================================================== *go to the page top

USAGE: m, w, z, isuppz, work, iwork, info, d, e, tryrac = NumRu::Lapack.cstemr( jobz, range, d, e, vl, vu, il, iu, nzc, tryrac, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CSTEMR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO ) * Purpose * ======= * * CSTEMR computes selected eigenvalues and, optionally, eigenvectors * of a real symmetric tridiagonal matrix T. Any such unreduced matrix has * a well defined set of pairwise different real eigenvalues, the corresponding * real eigenvectors are pairwise orthogonal. * * The spectrum may be computed either completely or partially by specifying * either an interval (VL,VU] or a range of indices IL:IU for the desired * eigenvalues. * * Depending on the number of desired eigenvalues, these are computed either * by bisection or the dqds algorithm. Numerically orthogonal eigenvectors are * computed by the use of various suitable L D L^T factorizations near clusters * of close eigenvalues (referred to as RRRs, Relatively Robust * Representations). An informal sketch of the algorithm 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. * * For more details, see: * - 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. * * Further Details * 1.CSTEMR works only on machines which follow IEEE-754 * floating-point standard in their handling of infinities and NaNs. * This permits the use of efficient inner loops avoiding a check for * zero divisors. * * 2. LAPACK routines can be used to reduce a complex Hermitean matrix to * real symmetric tridiagonal form. * * (Any complex Hermitean tridiagonal matrix has real values on its diagonal * and potentially complex numbers on its off-diagonals. By applying a * similarity transform with an appropriate diagonal matrix * diag(1,e^{i \phy_1}, ... , e^{i \phy_{n-1}}), the complex Hermitean * matrix can be transformed into a real symmetric matrix and complex * arithmetic can be entirely avoided.) * * While the eigenvectors of the real symmetric tridiagonal matrix are real, * the eigenvectors of original complex Hermitean matrix have complex entries * in general. * Since LAPACK drivers overwrite the matrix data with the eigenvectors, * CSTEMR accepts complex workspace to facilitate interoperability * with CUNMTR or CUPMTR. * * 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. * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the N diagonal elements of the tridiagonal matrix * T. On exit, D is overwritten. * * E (input/output) REAL array, dimension (N) * On entry, the (N-1) subdiagonal elements of the tridiagonal * matrix T in elements 1 to N-1 of E. E(N) need not be set on * input, but is used internally as workspace. * On exit, E is overwritten. * * VL (input) REAL * VU (input) REAL * 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. * Not referenced if RANGE = 'A' or 'V'. * * 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) REAL array, dimension (N) * The first M elements contain the selected eigenvalues in * ascending order. * * Z (output) COMPLEX array, dimension (LDZ, max(1,M) ) * If JOBZ = 'V', and if INFO = 0, then the first M columns of Z * contain the orthonormal eigenvectors of the matrix T * 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 can be computed with a workspace * query by setting NZC = -1, see below. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', then LDZ >= max(1,N). * * NZC (input) INTEGER * The number of eigenvectors to be held in the array Z. * If RANGE = 'A', then NZC >= max(1,N). * If RANGE = 'V', then NZC >= the number of eigenvalues in (VL,VU]. * If RANGE = 'I', then NZC >= IU-IL+1. * If NZC = -1, then a workspace query is assumed; the * routine calculates the number of columns of the array Z that * are needed to hold the eigenvectors. * This value is returned as the first entry of the Z array, and * no error message related to NZC is issued by XERBLA. * * 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 computed eigenvector * is nonzero only in elements ISUPPZ( 2*i-1 ) through * ISUPPZ( 2*i ). This is relevant in the case when the matrix * is split. ISUPPZ is only accessed when JOBZ is 'V' and N > 0. * * TRYRAC (input/output) LOGICAL * If TRYRAC.EQ..TRUE., indicates that the code should check whether * the tridiagonal matrix defines its eigenvalues to high relative * accuracy. If so, the code uses relative-accuracy preserving * algorithms that might be (a bit) slower depending on the matrix. * If the matrix does not define its eigenvalues to high relative * accuracy, the code can uses possibly faster algorithms. * If TRYRAC.EQ..FALSE., the code is not required to guarantee * relatively accurate eigenvalues and can use the fastest possible * techniques. * On exit, a .TRUE. TRYRAC will be set to .FALSE. if the matrix * does not define its eigenvalues to high relative accuracy. * * WORK (workspace/output) REAL array, dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal * (and minimal) LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= max(1,18*N) * if JOBZ = 'V', and LWORK >= max(1,12*N) if JOBZ = 'N'. * 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 (workspace/output) INTEGER array, dimension (LIWORK) * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of the array IWORK. LIWORK >= max(1,10*N) * if the eigenvectors are desired, and LIWORK >= max(1,8*N) * if only the eigenvalues are to be computed. * 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 (output) INTEGER * On exit, INFO * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = 1X, internal error in SLARRE, * if INFO = 2X, internal error in CLARRV. * Here, the digit X = ABS( IINFO ) < 10, where IINFO is * the nonzero error code returned by SLARRE or * CLARRV, respectively. * * * Further Details * =============== * * Based on contributions by * Beresford Parlett, University of California, Berkeley, USA * Jim Demmel, University of California, Berkeley, USA * Inderjit Dhillon, University of Texas, Austin, USA * Osni Marques, LBNL/NERSC, USA * Christof Voemel, University of California, Berkeley, USA * * ===================================================================== *go to the page top

USAGE: info, d, e, z = NumRu::Lapack.csteqr( compz, d, e, z, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE CSTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO ) * Purpose * ======= * * CSTEQR computes all eigenvalues and, optionally, eigenvectors of a * symmetric tridiagonal matrix using the implicit QL or QR method. * The eigenvectors of a full or band complex Hermitian matrix can also * be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this * matrix to tridiagonal form. * * Arguments * ========= * * COMPZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only. * = 'V': Compute eigenvalues and eigenvectors of the original * Hermitian matrix. On entry, Z must contain the * unitary matrix used to reduce the original matrix * to tridiagonal form. * = 'I': Compute eigenvalues and eigenvectors of the * tridiagonal matrix. Z is initialized to the identity * matrix. * * N (input) INTEGER * The order of the matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the diagonal elements of the tridiagonal matrix. * On exit, if INFO = 0, the eigenvalues in ascending order. * * E (input/output) REAL array, dimension (N-1) * On entry, the (n-1) subdiagonal elements of the tridiagonal * matrix. * On exit, E has been destroyed. * * Z (input/output) COMPLEX array, dimension (LDZ, N) * On entry, if COMPZ = 'V', then Z contains the unitary * matrix used in the reduction to tridiagonal form. * On exit, if INFO = 0, then if COMPZ = 'V', Z contains the * orthonormal eigenvectors of the original Hermitian matrix, * and if COMPZ = 'I', Z contains the orthonormal eigenvectors * of the symmetric tridiagonal matrix. * If COMPZ = 'N', then Z is not referenced. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * eigenvectors are desired, then LDZ >= max(1,N). * * WORK (workspace) REAL array, dimension (max(1,2*N-2)) * If COMPZ = 'N', then WORK is not referenced. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: the algorithm has failed to find all the eigenvalues in * a total of 30*N iterations; if INFO = i, then i * elements of E have not converged to zero; on exit, D * and E contain the elements of a symmetric tridiagonal * matrix which is unitarily similar to the original * matrix. * * ===================================================================== *go to the page top

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