Provided by: liblapack-doc_3.7.1-4ubuntu1_all

**NAME**

doubleOTHEReigen

**SYNOPSIS**

Functionssubroutinedbdsvdx(UPLO, JOBZ, RANGE, N, D, E, VL, VU, IL, IU, NS, S, Z, LDZ, WORK, IWORK, INFO)DBDSVDXsubroutinedggglm(N, M, P, A, LDA, B, LDB, D, X, Y, WORK, LWORK, INFO)DGGGLMsubroutinedsbev(JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO)DSBEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbev_2stage(JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, INFO)DSBEV_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbevd(JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSBEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbevd_2stage(JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSBEVD_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbevx(JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)DSBEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbevx_2stage(JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)DSBEVX_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedsbgv(JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO)DSBGVsubroutinedsbgvd(JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSBGVDsubroutinedsbgvx(JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)DSBGVXsubroutinedspev(JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO)DSPEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedspevd(JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSPEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedspevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)DSPEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedspgv(ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO)DSPGVsubroutinedspgvd(ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSPGVDsubroutinedspgvx(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)DSPGVXsubroutinedstev(JOBZ, N, D, E, Z, LDZ, WORK, INFO)DSTEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedstevd(JOBZ, N, D, E, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO)DSTEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedstevr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO)DSTEVRcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricessubroutinedstevx(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)DSTEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatrices

**Detailed** **Description**

This is the group of double Other Eigenvalue routines

**Function** **Documentation**

subroutinedbdsvdx(characterUPLO,characterJOBZ,characterRANGE,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,integerNS,doubleprecision,dimension(*)S,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integerINFO)DBDSVDXPurpose:DBDSVDX computes the singular value decomposition (SVD) of a real N-by-N (upper or lower) bidiagonal matrix B, B = U * S * VT, where S is a diagonal matrix with non-negative diagonal elements (the singular values of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively. Given an upper bidiagonal B with diagonal D = [ d_1 d_2 ... d_N ] and superdiagonal E = [ e_1 e_2 ... e_N-1 ], DBDSVDX computes the singular value decompositon of B through the eigenvalues and eigenvectors of the N*2-by-N*2 tridiagonal matrix | 0 d_1 | | d_1 0 e_1 | TGK = | e_1 0 d_2 | | d_2 . . | | . . . | If (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (+/-s,q), ||q|| = 1, are eigenpairs of TGK, with q = P * ( u' +/-v' ) / sqrt(2) = ( v_1 u_1 v_2 u_2 ... v_n u_n ) / sqrt(2), and P = [ e_{n+1} e_{1} e_{n+2} e_{2} ... ]. Given a TGK matrix, one can either a) compute -s,-v and change signs so that the singular values (and corresponding vectors) are already in descending order (as in DGESVD/DGESDD) or b) compute s,v and reorder the values (and corresponding vectors). DBDSVDX implements a) by calling DSTEVX (bisection plus inverse iteration, to be replaced with a version of the Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the MR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)Parameters:UPLOUPLO is CHARACTER*1 = 'U': B is upper bidiagonal; = 'L': B is lower bidiagonal.JOBZJOBZ is CHARACTER*1 = 'N': Compute singular values only; = 'V': Compute singular values and singular vectors.RANGERANGE is CHARACTER*1 = 'A': all singular values will be found. = 'V': all singular values in the half-open interval [VL,VU) will be found. = 'I': the IL-th through IU-th singular values will be found.NN is INTEGER The order of the bidiagonal matrix. N >= 0.DD is DOUBLE PRECISION array, dimension (N) The n diagonal elements of the bidiagonal matrix B.EE is DOUBLE PRECISION array, dimension (max(1,N-1)) The (n-1) superdiagonal elements of the bidiagonal matrix B in elements 1 to N-1.VLVL is DOUBLE PRECISION If RANGE='V', the lower bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.VUVU is DOUBLE PRECISION If RANGE='V', the upper bound of the interval to be searched for singular values. VU > VL. Not referenced if RANGE = 'A' or 'I'.ILIL is INTEGER If RANGE='I', the index of the smallest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.IUIU is INTEGER If RANGE='I', the index of the largest singular value to be returned. 1 <= IL <= IU <= min(M,N), if min(M,N) > 0. Not referenced if RANGE = 'A' or 'V'.NSNS is INTEGER The total number of singular values found. 0 <= NS <= N. If RANGE = 'A', NS = N, and if RANGE = 'I', NS = IU-IL+1.SS is DOUBLE PRECISION array, dimension (N) The first NS elements contain the selected singular values in ascending order.ZZ is DOUBLE PRECISION array, dimension (2*N,K) ) If JOBZ = 'V', then if INFO = 0 the first NS columns of Z contain the singular vectors of the matrix B corresponding to the selected singular values, with U in rows 1 to N and V in rows N+1 to N*2, i.e. Z = [ U ] [ V ] If JOBZ = 'N', then Z is not referenced. Note: The user must ensure that at least K = NS+1 columns are supplied in the array Z; if RANGE = 'V', the exact value of NS is not known in advance and an upper bound must be used.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(2,N*2).WORKWORK is DOUBLE PRECISION array, dimension (14*N)IWORKIWORK is INTEGER array, dimension (12*N) If JOBZ = 'V', then if INFO = 0, the first NS elements of IWORK are zero. If INFO > 0, then IWORK contains the indices of the eigenvectors that failed to converge in DSTEVX.INFOINFO is 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 DSTEVX. The indices of the eigenvectors (as returned by DSTEVX) are stored in the array IWORK. if INFO = N*2 + 1, an internal error occurred.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinedggglm(integerN,integerM,integerP,doubleprecision,dimension(lda,*)A,integerLDA,doubleprecision,dimension(ldb,*)B,integerLDB,doubleprecision,dimension(*)D,doubleprecision,dimension(*)X,doubleprecision,dimension(*)Y,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DGGGLMPurpose:DGGGLM solves a general Gauss-Markov linear model (GLM) problem: minimize || y ||_2 subject to d = A*x + B*y x where A is an N-by-M matrix, B is an N-by-P matrix, and d is a given N-vector. It is assumed that M <= N <= M+P, and rank(A) = M and rank( A B ) = N. Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B) given by A = Q*(R), B = Q*T*Z. (0) In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem minimize || inv(B)*(d-A*x) ||_2 x where inv(B) denotes the inverse of B.Parameters:NN is INTEGER The number of rows of the matrices A and B. N >= 0.MM is INTEGER The number of columns of the matrix A. 0 <= M <= N.PP is INTEGER The number of columns of the matrix B. P >= N-M.AA is DOUBLE PRECISION array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the upper triangular part of the array A contains the M-by-M upper triangular matrix R.LDALDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).BB is DOUBLE PRECISION array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)th subdiagonal contain the N-by-P upper trapezoidal matrix T.LDBLDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).DD is DOUBLE PRECISION array, dimension (N) On entry, D is the left hand side of the GLM equation. On exit, D is destroyed.XX is DOUBLE PRECISION array, dimension (M)YY is DOUBLE PRECISION array, dimension (P) On exit, X and Y are the solutions of the GLM problem.WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N+M+P). For optimum performance, LWORK >= M+min(N,P)+max(N,P)*NB, where NB is an upper bound for the optimal blocksizes for DGEQRF, SGERQF, DORMQR and SORMRQ. 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.INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. = 1: the upper triangular factor R associated with A in the generalized QR factorization of the pair (A, B) is singular, so that rank(A) < M; the least squares solution could not be computed. = 2: the bottom (N-M) by (N-M) part of the upper trapezoidal factor T associated with B in the generalized QR factorization of the pair (A, B) is singular, so that rank( A B ) < N; the least squares solution could not be computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsbev(characterJOBZ,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerINFO)DSBEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (max(1,3*N-2))INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsbev_2stage(characterJOBZ,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integerINFO)DSBEV_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEV_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension LWORK On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS + N where KD is the size of the band. 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017FurtherDetails: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/28/2/196subroutinedsbevd(characterJOBZ,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSBEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. IF N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 2, LWORK must be at least 2*N. If JOBZ = 'V' and N > 2, LWORK must be at least ( 1 + 5*N + 2*N**2 ). 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsbevd_2stage(characterJOBZ,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSBEVD_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEVD_2STAGE computes all the eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A using the 2stage technique for the reduction to tridiagonal. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension LWORK On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, dimension) where dimension = (2KD+1)*N + KD*NTHREADS + N where KD is the size of the band. 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 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 2, LIWORK must be at least 3 + 5*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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016FurtherDetails: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/28/2/196subroutinedsbevx(characterJOBZ,characterRANGE,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(ldq,*)Q,integerLDQ,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSBEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.QQ is DOUBLE PRECISION array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQLDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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 AB to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.ZZ 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (7*N)IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is 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. Their indices are stored in array IFAIL.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinedsbevx_2stage(characterJOBZ,characterRANGE,characterUPLO,integerN,integerKD,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(ldq,*)Q,integerLDQ,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSBEVX_2STAGEcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSBEVX_2STAGE computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors. Not available in this release.RANGERANGE 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.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.KDKD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KD >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). On exit, AB is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of AB.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KD + 1.QQ is DOUBLE PRECISION array, dimension (LDQ, N) If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQLDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'V', then LDQ >= max(1,N).VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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 AB to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.ZZ 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (LWORK)LWORKLWORK is INTEGER The length of the array WORK. LWORK >= 1, when N <= 1; otherwise If JOBZ = 'N' and N > 1, LWORK must be queried. LWORK = MAX(1, 7*N, dimension) where dimension = (2KD+1)*N + KD*NTHREADS + 2*N where KD is the size of the band. 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.IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is 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. Their indices are stored in array IFAIL.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016FurtherDetails: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/28/2/196subroutinedsbgv(characterJOBZ,characterUPLO,integerN,integerKA,integerKB,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(ldbb,*)BB,integerLDBB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerINFO)DSBGVPurpose:DSBGV computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.NN is INTEGER The order of the matrices A and B. N >= 0.KAKA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KBKB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BBBB is DOUBLE PRECISION array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.LDBBLDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so that Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= N.WORKWORK is DOUBLE PRECISION array, dimension (3*N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedsbgvd(characterJOBZ,characterUPLO,integerN,integerKA,integerKB,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(ldbb,*)BB,integerLDBB,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSBGVDPurpose:DSBGVD computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.NN is INTEGER The order of the matrices A and B. N >= 0.KAKA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KBKB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BBBB is DOUBLE PRECISION array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.LDBBLDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, and i is: <= N: the algorithm failed to converge: i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF returned INFO = i: B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Mark Fahey, Department of Mathematics, Univ. of Kentucky, USAsubroutinedsbgvx(characterJOBZ,characterRANGE,characterUPLO,integerN,integerKA,integerKB,doubleprecision,dimension(ldab,*)AB,integerLDAB,doubleprecision,dimension(ldbb,*)BB,integerLDBB,doubleprecision,dimension(ldq,*)Q,integerLDQ,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSBGVXPurpose:DSBGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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.UPLOUPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.NN is INTEGER The order of the matrices A and B. N >= 0.KAKA is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA >= 0.KBKB is INTEGER The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB >= 0.ABAB is DOUBLE PRECISION array, dimension (LDAB, N) On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). On exit, the contents of AB are destroyed.LDABLDAB is INTEGER The leading dimension of the array AB. LDAB >= KA+1.BBBB is DOUBLE PRECISION array, dimension (LDBB, N) On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by DPBSTF.LDBBLDBB is INTEGER The leading dimension of the array BB. LDBB >= KB+1.QQ is DOUBLE PRECISION array, dimension (LDQ, N) If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.LDQLDQ is INTEGER The leading dimension of the array Q. If JOBZ = 'N', LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (7*N)IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (M) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is INTEGER = 0 : successful exit < 0 : if INFO = -i, the i-th argument had an illegal value <= N: if INFO = i, then i eigenvectors failed to converge. Their indices are stored in IFAIL. > N : DPBSTF returned an error code; i.e., if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Mark Fahey, Department of Mathematics, Univ. of Kentucky, USAsubroutinedspev(characterJOBZ,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerINFO)DSPEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSPEV computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (3*N)INFOINFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedspevd(characterJOBZ,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSPEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSPEVD computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with W(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the required LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. If N <= 1, LWORK must be at least 1. If JOBZ = 'N' and N > 1, LWORK must be at least 2*N. If JOBZ = 'V' and N > 1, LWORK must be at least 1 + 6*N + N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the required 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the required LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK must be at least 1. If JOBZ = 'V' and N > 1, LIWORK must be at least 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017subroutinedspevx(characterJOBZ,characterRANGE,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSPEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSPEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in packed storage. Eigenvalues/vectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.NN is INTEGER The order of the matrix A. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, AP is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the diagonal and first superdiagonal of the tridiagonal matrix T overwrite the corresponding elements of A, and if UPLO = 'L', the diagonal and first subdiagonal of T overwrite the corresponding elements of A.VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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 AP to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the selected eigenvalues in ascending order.ZZ 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 an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (8*N)IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is 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. Their indices are stored in array IFAIL.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016subroutinedspgv(integerITYPE,characterJOBZ,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)BP,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerINFO)DSPGVPurpose:DSPGV computes all the eigenvalues and, optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.Parameters:ITYPEITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.NN is INTEGER The order of the matrices A and B. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BPBP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (3*N)INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: DPPTRF or DSPEV returned an error code: <= N: if INFO = i, DSPEV failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero. > N: if INFO = n + i, for 1 <= i <= n, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2017subroutinedspgvd(integerITYPE,characterJOBZ,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)BP,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSPGVDPurpose:DSPGVD computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed format, and B is also positive definite. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:ITYPEITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.UPLOUPLO is CHARACTER*1 = 'U': Upper triangles of A and B are stored; = 'L': Lower triangles of A and B are stored.NN is INTEGER The order of the matrices A and B. N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BPBP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.WW is DOUBLE PRECISION array, dimension (N) If INFO = 0, the eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors. The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the required LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. If N <= 1, LWORK >= 1. If JOBZ = 'N' and N > 1, LWORK >= 2*N. If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2. If LWORK = -1, then a workspace query is assumed; the routine only calculates the required 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the required LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1, LIWORK >= 1. If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. If LIWORK = -1, then a workspace query is assumed; the routine only calculates the required 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: DPPTRF or DSPEVD returned an error code: <= N: if INFO = i, DSPEVD failed to converge; i off-diagonal elements of an intermediate tridiagonal form did not converge to zero; > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016Contributors:Mark Fahey, Department of Mathematics, Univ. of Kentucky, USAsubroutinedspgvx(integerITYPE,characterJOBZ,characterRANGE,characterUPLO,integerN,doubleprecision,dimension(*)AP,doubleprecision,dimension(*)BP,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSPGVXPurpose:DSPGVX computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric, stored in packed storage, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.Parameters:ITYPEITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*xJOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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.UPLOUPLO is CHARACTER*1 = 'U': Upper triangle of A and B are stored; = 'L': Lower triangle of A and B are stored.NN is INTEGER The order of the matrix pencil (A,B). N >= 0.APAP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.BPBP is DOUBLE PRECISION array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S').MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.ZZ is DOUBLE PRECISION array, dimension (LDZ, max(1,M)) If JOBZ = 'N', then Z is not referenced. 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). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (8*N)IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: DPPTRF or DSPEVX returned an error code: <= N: if INFO = i, DSPEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Mark Fahey, Department of Mathematics, Univ. of Kentucky, USAsubroutinedstev(characterJOBZ,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerINFO)DSTEVcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSTEV computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.NN is INTEGER The order of the matrix. N >= 0.DD is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, if INFO = 0, the eigenvalues in ascending order.EE is DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to N-1 of E. On exit, the contents of E are destroyed.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with D(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (max(1,2*N-2)) If JOBZ = 'N', WORK is not referenced.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of E did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedstevd(characterJOBZ,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSTEVDcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSTEVD computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix. If eigenvectors are desired, it uses a divide and conquer algorithm. The divide and conquer algorithm 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.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.NN is INTEGER The order of the matrix. N >= 0.DD is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, if INFO = 0, the eigenvalues in ascending order.EE is DOUBLE PRECISION array, dimension (N-1) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A, stored in elements 1 to N-1 of E. On exit, the contents of E are destroyed.ZZ is DOUBLE PRECISION array, dimension (LDZ, N) If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal eigenvectors of the matrix A, with the i-th column of Z holding the eigenvector associated with D(i). If JOBZ = 'N', then Z is not referenced.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. If JOBZ = 'N' or N <= 1 then LWORK must be at least 1. If JOBZ = 'V' and N > 1 then LWORK must be at least ( 1 + 4*N + N**2 ). 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.LIWORKLIWORK is INTEGER The dimension of the array IWORK. If JOBZ = 'N' or N <= 1 then LIWORK must be at least 1. If JOBZ = 'V' and N > 1 then LIWORK must be at least 3+5*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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, the algorithm failed to converge; i off-diagonal elements of E did not converge to zero.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:December 2016subroutinedstevr(characterJOBZ,characterRANGE,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,integer,dimension(*)ISUPPZ,doubleprecision,dimension(*)WORK,integerLWORK,integer,dimension(*)IWORK,integerLIWORK,integerINFO)DSTEVRcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. Whenever possible, DSTEVR 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 the i-th unreduced block of T, (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T is a relatively robust representation, (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high relative accuracy by the dqds algorithm, (c) If there is a cluster of close eigenvalues, "choose" sigma_i close to the cluster, and go to step (a), (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T, compute the corresponding eigenvector by forming a rank-revealing twisted factorization. The desired accuracy of the output can be specified by the input parameter ABSTOL. For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997. Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. DSTEVR 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:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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 calledNN is INTEGER The order of the matrix. N >= 0.DD is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.EE is DOUBLE PRECISION array, dimension (max(1,N-1)) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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.MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.ZZ 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). 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).ISUPPZISUPPZ 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 ). Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1WORKWORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal (and minimal) LWORK.LWORKLWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,20*N). 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.IWORKIWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal (and minimal) LIWORK.LIWORKLIWORK 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 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.INFOINFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal errorAuthor:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016Contributors:Inderjit Dhillon, IBM Almaden, USA Osni Marques, LBNL/NERSC, USA Ken Stanley, Computer Science Division, University of California at Berkeley, USAsubroutinedstevx(characterJOBZ,characterRANGE,integerN,doubleprecision,dimension(*)D,doubleprecision,dimension(*)E,doubleprecisionVL,doubleprecisionVU,integerIL,integerIU,doubleprecisionABSTOL,integerM,doubleprecision,dimension(*)W,doubleprecision,dimension(ldz,*)Z,integerLDZ,doubleprecision,dimension(*)WORK,integer,dimension(*)IWORK,integer,dimension(*)IFAIL,integerINFO)DSTEVXcomputestheeigenvaluesand,optionally,theleftand/orrighteigenvectorsforOTHERmatricesPurpose:DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.Parameters:JOBZJOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.RANGERANGE 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.NN is INTEGER The order of the matrix. N >= 0.DD is DOUBLE PRECISION array, dimension (N) On entry, the n diagonal elements of the tridiagonal matrix A. On exit, D may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.EE is DOUBLE PRECISION array, dimension (max(1,N-1)) On entry, the (n-1) subdiagonal elements of the tridiagonal matrix A in elements 1 to N-1 of E. On exit, E may be multiplied by a constant factor chosen to avoid over/underflow in computing the eigenvalues.VLVL 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'.VUVU 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'.ILIL 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'.IUIU 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'.ABSTOLABSTOL 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. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.MM is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.WW is DOUBLE PRECISION array, dimension (N) The first M elements contain the selected eigenvalues in ascending order.ZZ 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 an eigenvector fails to converge (INFO > 0), then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. 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.LDZLDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).WORKWORK is DOUBLE PRECISION array, dimension (5*N)IWORKIWORK is INTEGER array, dimension (5*N)IFAILIFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.INFOINFO is 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. Their indices are stored in array IFAIL.Author:Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.Date:June 2016

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