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---
:name: dsbevx
:md5sum: c54c0d69a5f4df0fc356205a4641b5ae
:category: :subroutine
:arguments:
- jobz:
:type: char
:intent: input
- range:
:type: char
:intent: input
- uplo:
:type: char
:intent: input
- n:
:type: integer
:intent: input
- kd:
:type: integer
:intent: input
- ab:
:type: doublereal
:intent: input/output
:dims:
- ldab
- n
- ldab:
:type: integer
:intent: input
- q:
:type: doublereal
:intent: output
:dims:
- ldq
- n
- ldq:
:type: integer
:intent: input
- vl:
:type: doublereal
:intent: input
- vu:
:type: doublereal
:intent: input
- il:
:type: integer
:intent: input
- iu:
:type: integer
:intent: input
- abstol:
:type: doublereal
:intent: input
- m:
:type: integer
:intent: output
- w:
:type: doublereal
:intent: output
:dims:
- n
- z:
:type: doublereal
:intent: output
:dims:
- ldz
- MAX(1,m)
- ldz:
:type: integer
:intent: input
- work:
:type: doublereal
:intent: workspace
:dims:
- 7*n
- iwork:
:type: integer
:intent: workspace
:dims:
- 5*n
- ifail:
:type: integer
:intent: output
:dims:
- n
- info:
:type: integer
:intent: output
:substitutions:
ldz: "lsame_(&jobz,\"V\") ? MAX(1,n) : 1"
m: "lsame_(&range,\"A\") ? n : lsame_(&range,\"I\") ? iu-il+1 : 0"
ldq: "lsame_(&jobz,\"V\") ? MAX(1,n) : 0"
:fortran_help: " SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DSBEVX computes selected eigenvalues and, optionally, eigenvectors\n\
* of a real symmetric band matrix A. Eigenvalues and eigenvectors can\n\
* be selected by specifying either a range of values or a range of\n\
* indices for the desired eigenvalues.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* JOBZ (input) CHARACTER*1\n\
* = 'N': Compute eigenvalues only;\n\
* = 'V': Compute eigenvalues and eigenvectors.\n\
*\n\
* RANGE (input) CHARACTER*1\n\
* = 'A': all eigenvalues will be found;\n\
* = 'V': all eigenvalues in the half-open interval (VL,VU]\n\
* will be found;\n\
* = 'I': the IL-th through IU-th eigenvalues will be found.\n\
*\n\
* UPLO (input) CHARACTER*1\n\
* = 'U': Upper triangle of A is stored;\n\
* = 'L': Lower triangle of A is stored.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A. N >= 0.\n\
*\n\
* KD (input) INTEGER\n\
* The number of superdiagonals of the matrix A if UPLO = 'U',\n\
* or the number of subdiagonals if UPLO = 'L'. KD >= 0.\n\
*\n\
* AB (input/output) DOUBLE PRECISION array, dimension (LDAB, N)\n\
* On entry, the upper or lower triangle of the symmetric band\n\
* matrix A, stored in the first KD+1 rows of the array. The\n\
* j-th column of A is stored in the j-th column of the array AB\n\
* as follows:\n\
* if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;\n\
* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).\n\
*\n\
* On exit, AB is overwritten by values generated during the\n\
* reduction to tridiagonal form. If UPLO = 'U', the first\n\
* superdiagonal and the diagonal of the tridiagonal matrix T\n\
* are returned in rows KD and KD+1 of AB, and if UPLO = 'L',\n\
* the diagonal and first subdiagonal of T are returned in the\n\
* first two rows of AB.\n\
*\n\
* LDAB (input) INTEGER\n\
* The leading dimension of the array AB. LDAB >= KD + 1.\n\
*\n\
* Q (output) DOUBLE PRECISION array, dimension (LDQ, N)\n\
* If JOBZ = 'V', the N-by-N orthogonal matrix used in the\n\
* reduction to tridiagonal form.\n\
* If JOBZ = 'N', the array Q is not referenced.\n\
*\n\
* LDQ (input) INTEGER\n\
* The leading dimension of the array Q. If JOBZ = 'V', then\n\
* LDQ >= max(1,N).\n\
*\n\
* VL (input) DOUBLE PRECISION\n\
* VU (input) DOUBLE PRECISION\n\
* If RANGE='V', the lower and upper bounds of the interval to\n\
* be searched for eigenvalues. VL < VU.\n\
* Not referenced if RANGE = 'A' or 'I'.\n\
*\n\
* IL (input) INTEGER\n\
* IU (input) INTEGER\n\
* If RANGE='I', the indices (in ascending order) of the\n\
* smallest and largest eigenvalues to be returned.\n\
* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.\n\
* Not referenced if RANGE = 'A' or 'V'.\n\
*\n\
* ABSTOL (input) DOUBLE PRECISION\n\
* The absolute error tolerance for the eigenvalues.\n\
* An approximate eigenvalue is accepted as converged\n\
* when it is determined to lie in an interval [a,b]\n\
* of width less than or equal to\n\
*\n\
* ABSTOL + EPS * max( |a|,|b| ) ,\n\
*\n\
* where EPS is the machine precision. If ABSTOL is less than\n\
* or equal to zero, then EPS*|T| will be used in its place,\n\
* where |T| is the 1-norm of the tridiagonal matrix obtained\n\
* by reducing AB to tridiagonal form.\n\
*\n\
* Eigenvalues will be computed most accurately when ABSTOL is\n\
* set to twice the underflow threshold 2*DLAMCH('S'), not zero.\n\
* If this routine returns with INFO>0, indicating that some\n\
* eigenvectors did not converge, try setting ABSTOL to\n\
* 2*DLAMCH('S').\n\
*\n\
* See \"Computing Small Singular Values of Bidiagonal Matrices\n\
* with Guaranteed High Relative Accuracy,\" by Demmel and\n\
* Kahan, LAPACK Working Note #3.\n\
*\n\
* M (output) INTEGER\n\
* The total number of eigenvalues found. 0 <= M <= N.\n\
* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.\n\
*\n\
* W (output) DOUBLE PRECISION array, dimension (N)\n\
* The first M elements contain the selected eigenvalues in\n\
* ascending order.\n\
*\n\
* Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))\n\
* If JOBZ = 'V', then if INFO = 0, the first M columns of Z\n\
* contain the orthonormal eigenvectors of the matrix A\n\
* corresponding to the selected eigenvalues, with the i-th\n\
* column of Z holding the eigenvector associated with W(i).\n\
* If an eigenvector fails to converge, then that column of Z\n\
* contains the latest approximation to the eigenvector, and the\n\
* index of the eigenvector is returned in IFAIL.\n\
* If JOBZ = 'N', then Z is not referenced.\n\
* Note: the user must ensure that at least max(1,M) columns are\n\
* supplied in the array Z; if RANGE = 'V', the exact value of M\n\
* is not known in advance and an upper bound must be used.\n\
*\n\
* LDZ (input) INTEGER\n\
* The leading dimension of the array Z. LDZ >= 1, and if\n\
* JOBZ = 'V', LDZ >= max(1,N).\n\
*\n\
* WORK (workspace) DOUBLE PRECISION array, dimension (7*N)\n\
*\n\
* IWORK (workspace) INTEGER array, dimension (5*N)\n\
*\n\
* IFAIL (output) INTEGER array, dimension (N)\n\
* If JOBZ = 'V', then if INFO = 0, the first M elements of\n\
* IFAIL are zero. If INFO > 0, then IFAIL contains the\n\
* indices of the eigenvectors that failed to converge.\n\
* If JOBZ = 'N', then IFAIL is not referenced.\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit.\n\
* < 0: if INFO = -i, the i-th argument had an illegal value.\n\
* > 0: if INFO = i, then i eigenvectors failed to converge.\n\
* Their indices are stored in array IFAIL.\n\
*\n\n\
* =====================================================================\n\
*\n"
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