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version 1.19, 2023/08/07 08:39:05
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*> \brief \b DSBGVD |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download DSBGVD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsbgvd.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsbgvd.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsbgvd.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, |
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* Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBZ, UPLO |
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* INTEGER INFO, KA, KB, LDAB, LDBB, LDZ, LIWORK, LWORK, N |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IWORK( * ) |
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* DOUBLE PRECISION AB( LDAB, * ), BB( LDBB, * ), W( * ), |
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* $ WORK( * ), Z( LDZ, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DSBGVD computes all the eigenvalues, and optionally, the eigenvectors |
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*> of a real generalized symmetric-definite banded eigenproblem, of the |
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*> form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and |
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*> banded, and B is also positive definite. If eigenvectors are |
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*> desired, it uses a divide and conquer algorithm. |
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*> |
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*> The divide and conquer algorithm makes very mild assumptions about |
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*> floating point arithmetic. It will work on machines with a guard |
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*> digit in add/subtract, or on those binary machines without guard |
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*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or |
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*> Cray-2. It could conceivably fail on hexadecimal or decimal machines |
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*> without guard digits, but we know of none. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] JOBZ |
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*> \verbatim |
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*> JOBZ is CHARACTER*1 |
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*> = 'N': Compute eigenvalues only; |
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*> = 'V': Compute eigenvalues and eigenvectors. |
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*> \endverbatim |
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*> |
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*> \param[in] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> = 'U': Upper triangles of A and B are stored; |
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*> = 'L': Lower triangles of A and B are stored. |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KA |
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*> \verbatim |
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*> KA is INTEGER |
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*> The number of superdiagonals of the matrix A if UPLO = 'U', |
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*> or the number of subdiagonals if UPLO = 'L'. KA >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KB |
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*> \verbatim |
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*> KB is INTEGER |
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*> The number of superdiagonals of the matrix B if UPLO = 'U', |
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*> or the number of subdiagonals if UPLO = 'L'. KB >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] AB |
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*> \verbatim |
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*> AB is DOUBLE PRECISION array, dimension (LDAB, N) |
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*> On entry, the upper or lower triangle of the symmetric band |
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*> matrix A, stored in the first ka+1 rows of the array. The |
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*> j-th column of A is stored in the j-th column of the array AB |
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*> as follows: |
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*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; |
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). |
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*> |
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*> On exit, the contents of AB are destroyed. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAB |
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*> \verbatim |
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*> LDAB is INTEGER |
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*> The leading dimension of the array AB. LDAB >= KA+1. |
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*> \endverbatim |
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*> |
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*> \param[in,out] BB |
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*> \verbatim |
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*> BB is DOUBLE PRECISION array, dimension (LDBB, N) |
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*> On entry, the upper or lower triangle of the symmetric band |
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*> matrix B, stored in the first kb+1 rows of the array. The |
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*> j-th column of B is stored in the j-th column of the array BB |
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*> as follows: |
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*> if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; |
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*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). |
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*> |
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*> On exit, the factor S from the split Cholesky factorization |
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*> B = S**T*S, as returned by DPBSTF. |
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*> \endverbatim |
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*> |
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*> \param[in] LDBB |
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*> \verbatim |
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*> LDBB is INTEGER |
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*> The leading dimension of the array BB. LDBB >= KB+1. |
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*> \endverbatim |
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*> |
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*> \param[out] W |
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*> \verbatim |
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*> W is DOUBLE PRECISION array, dimension (N) |
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*> If INFO = 0, the eigenvalues in ascending order. |
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*> \endverbatim |
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*> |
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*> \param[out] Z |
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*> \verbatim |
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*> Z is DOUBLE PRECISION array, dimension (LDZ, N) |
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*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of |
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*> eigenvectors, with the i-th column of Z holding the |
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*> eigenvector associated with W(i). The eigenvectors are |
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*> normalized so Z**T*B*Z = I. |
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*> If JOBZ = 'N', then Z is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDZ |
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*> \verbatim |
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*> LDZ is INTEGER |
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*> The leading dimension of the array Z. LDZ >= 1, and if |
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*> JOBZ = 'V', LDZ >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
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*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LWORK |
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*> \verbatim |
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*> LWORK is INTEGER |
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*> The dimension of the array WORK. |
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*> If N <= 1, LWORK >= 1. |
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*> If JOBZ = 'N' and N > 1, LWORK >= 2*N. |
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*> If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. |
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*> |
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*> If LWORK = -1, then a workspace query is assumed; the routine |
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*> only calculates the optimal sizes of the WORK and IWORK |
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*> arrays, returns these values as the first entries of the WORK |
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*> and IWORK arrays, and no error message related to LWORK or |
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*> LIWORK is issued by XERBLA. |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (MAX(1,LIWORK)) |
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*> On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. |
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*> \endverbatim |
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*> |
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*> \param[in] LIWORK |
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*> \verbatim |
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*> LIWORK is INTEGER |
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*> The dimension of the array IWORK. |
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*> If JOBZ = 'N' or N <= 1, LIWORK >= 1. |
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*> If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. |
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*> |
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*> If LIWORK = -1, then a workspace query is assumed; the |
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*> routine only calculates the optimal sizes of the WORK and |
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*> IWORK arrays, returns these values as the first entries of |
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*> the WORK and IWORK arrays, and no error message related to |
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*> LWORK or LIWORK is issued by XERBLA. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, and i is: |
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*> <= N: the algorithm failed to converge: |
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*> i off-diagonal elements of an intermediate |
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*> tridiagonal form did not converge to zero; |
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*> > N: if INFO = N + i, for 1 <= i <= N, then DPBSTF |
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*> returned INFO = i: B is not positive definite. |
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*> The factorization of B could not be completed and |
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*> no eigenvalues or eigenvectors were computed. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \ingroup doubleOTHEReigen |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA |
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* |
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* ===================================================================== |
SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, |
SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, |
$ Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) |
$ Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
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* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBZ, UPLO |
CHARACTER JOBZ, UPLO |
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$ WORK( * ), Z( LDZ, * ) |
$ WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
|
* ======= |
|
* |
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* 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. |
|
* |
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* Arguments |
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* ========= |
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* |
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* JOBZ (input) CHARACTER*1 |
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* = 'N': Compute eigenvalues only; |
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* = 'V': Compute eigenvalues and eigenvectors. |
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* |
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* UPLO (input) CHARACTER*1 |
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* = 'U': Upper triangles of A and B are stored; |
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* = 'L': Lower triangles of A and B are stored. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A and B. N >= 0. |
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* |
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* KA (input) INTEGER |
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* The number of superdiagonals of the matrix A if UPLO = 'U', |
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* or the number of subdiagonals if UPLO = 'L'. KA >= 0. |
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* |
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* KB (input) INTEGER |
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* The number of superdiagonals of the matrix B if UPLO = 'U', |
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* or the number of subdiagonals if UPLO = 'L'. KB >= 0. |
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* |
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* AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N) |
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* On entry, the upper or lower triangle of the symmetric band |
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* matrix A, stored in the first ka+1 rows of the array. The |
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* j-th column of A is stored in the j-th column of the array AB |
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* as follows: |
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* if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; |
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* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). |
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* |
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* On exit, the contents of AB are destroyed. |
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* |
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* LDAB (input) INTEGER |
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* The leading dimension of the array AB. LDAB >= KA+1. |
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* |
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* BB (input/output) DOUBLE PRECISION array, dimension (LDBB, N) |
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* On entry, the upper or lower triangle of the symmetric band |
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* matrix B, stored in the first kb+1 rows of the array. The |
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* j-th column of B is stored in the j-th column of the array BB |
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* as follows: |
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* if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; |
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* if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). |
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* |
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* On exit, the factor S from the split Cholesky factorization |
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* B = S**T*S, as returned by DPBSTF. |
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* |
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* LDBB (input) INTEGER |
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* The leading dimension of the array BB. LDBB >= KB+1. |
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* |
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* W (output) DOUBLE PRECISION array, dimension (N) |
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* If INFO = 0, the eigenvalues in ascending order. |
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* |
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* Z (output) DOUBLE PRECISION array, dimension (LDZ, N) |
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* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of |
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* eigenvectors, with the i-th column of Z holding the |
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* eigenvector associated with W(i). The eigenvectors are |
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* normalized so Z**T*B*Z = I. |
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* If JOBZ = 'N', then Z is not referenced. |
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* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. LDZ >= 1, and if |
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* JOBZ = 'V', LDZ >= max(1,N). |
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* |
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* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) |
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. |
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* |
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* LWORK (input) INTEGER |
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* The dimension of the array WORK. |
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* If N <= 1, LWORK >= 1. |
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* If JOBZ = 'N' and N > 1, LWORK >= 3*N. |
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* If JOBZ = 'V' and N > 1, LWORK >= 1 + 5*N + 2*N**2. |
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* |
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* If LWORK = -1, then a workspace query is assumed; the routine |
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* only calculates the optimal sizes of the WORK and IWORK |
|
* arrays, returns these values as the first entries of the WORK |
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* and IWORK arrays, and no error message related to LWORK or |
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* LIWORK is issued by XERBLA. |
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* |
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* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) |
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* On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. |
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* |
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* LIWORK (input) INTEGER |
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* The dimension of the array IWORK. |
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* If JOBZ = 'N' or N <= 1, LIWORK >= 1. |
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* If JOBZ = 'V' and N > 1, LIWORK >= 3 + 5*N. |
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* |
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* If LIWORK = -1, then a workspace query is assumed; the |
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* routine only calculates the optimal sizes of the WORK and |
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* IWORK arrays, returns these values as the first entries of |
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* the WORK and IWORK arrays, and no error message related to |
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* LWORK or LIWORK is issued by XERBLA. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* > 0: if INFO = i, and i is: |
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* <= N: the algorithm failed to converge: |
|
* i off-diagonal elements of an intermediate |
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* 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. |
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* The factorization of B could not be completed and |
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* no eigenvalues or eigenvectors were computed. |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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INDWK2 = INDWRK + N*N |
INDWK2 = INDWRK + N*N |
LLWRK2 = LWORK - INDWK2 + 1 |
LLWRK2 = LWORK - INDWK2 + 1 |
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, |
CALL DSBGST( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Z, LDZ, |
$ WORK( INDWRK ), IINFO ) |
$ WORK, IINFO ) |
* |
* |
* Reduce to tridiagonal form. |
* Reduce to tridiagonal form. |
* |
* |