USAGE: w, z, info, ab = NumRu::Lapack.dsbev( jobz, uplo, kd, ab, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBEV( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, INFO ) * Purpose * ======= * * DSBEV computes all the eigenvalues and, optionally, eigenvectors of * a real symmetric band matrix A. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD + 1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace) DOUBLE PRECISION array, dimension (max(1,3*N-2)) * * INFO (output) 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. * * ===================================================================== *go to the page top

USAGE: w, z, work, iwork, info, ab = NumRu::Lapack.dsbevd( jobz, uplo, kd, ab, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBEVD( JOBZ, UPLO, N, KD, AB, LDAB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) * Purpose * ======= * * 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. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD + 1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace/output) DOUBLE PRECISION array, * dimension (LWORK) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) 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. * * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) INTEGER * The dimension of the array LIWORK. * 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. * * INFO (output) 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. * * ===================================================================== *go to the page top

USAGE: q, m, w, z, ifail, info, ab = NumRu::Lapack.dsbevx( jobz, range, uplo, kd, ab, vl, vu, il, iu, abstol, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO ) * Purpose * ======= * * 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. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * RANGE (input) CHARACTER*1 * = 'A': all eigenvalues will be found; * = 'V': all eigenvalues in the half-open interval (VL,VU] * will be found; * = 'I': the IL-th through IU-th eigenvalues will be found. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD + 1. * * Q (output) 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. * * LDQ (input) INTEGER * The leading dimension of the array Q. If JOBZ = 'V', then * LDQ >= max(1,N). * * VL (input) DOUBLE PRECISION * VU (input) DOUBLE PRECISION * If RANGE='V', the lower and upper bounds of the interval to * be searched for eigenvalues. VL < VU. * Not referenced if RANGE = 'A' or 'I'. * * IL (input) INTEGER * IU (input) INTEGER * If RANGE='I', the indices (in ascending order) of the * smallest and largest eigenvalues to be returned. * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. * Not referenced if RANGE = 'A' or 'V'. * * ABSTOL (input) DOUBLE PRECISION * The absolute error tolerance for the eigenvalues. * An approximate eigenvalue is accepted as converged * when it is determined to lie in an interval [a,b] * of width less than or equal to * * ABSTOL + EPS * max( |a|,|b| ) , * * where EPS is the machine precision. If ABSTOL is less than * or equal to zero, then EPS*|T| will be used in its place, * where |T| is the 1-norm of the tridiagonal matrix obtained * by reducing 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. * * M (output) INTEGER * The total number of eigenvalues found. 0 <= M <= N. * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. * * W (output) DOUBLE PRECISION array, dimension (N) * The first M elements contain the selected eigenvalues in * ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace) DOUBLE PRECISION array, dimension (7*N) * * IWORK (workspace) INTEGER array, dimension (5*N) * * IFAIL (output) 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. * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = i, then i eigenvectors failed to converge. * Their indices are stored in array IFAIL. * * ===================================================================== *go to the page top

USAGE: x, info, ab = NumRu::Lapack.dsbgst( vect, uplo, ka, kb, ab, bb, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBGST( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO ) * Purpose * ======= * * DSBGST reduces a real symmetric-definite banded generalized * eigenproblem A*x = lambda*B*x to standard form C*y = lambda*y, * such that C has the same bandwidth as A. * * B must have been previously factorized as S**T*S by DPBSTF, using a * split Cholesky factorization. A is overwritten by C = X**T*A*X, where * X = S**(-1)*Q and Q is an orthogonal matrix chosen to preserve the * bandwidth of A. * * Arguments * ========= * * VECT (input) CHARACTER*1 * = 'N': do not form the transformation matrix X; * = 'V': form X. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * KA (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= 0. * * KB (input) INTEGER * The number of superdiagonals of the matrix B if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= KB >= 0. * * AB (input/output) 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 transformed matrix X**T*A*X, stored in the same * format as A. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KA+1. * * BB (input) DOUBLE PRECISION array, dimension (LDBB,N) * The banded factor S from the split Cholesky factorization of * B, as returned by DPBSTF, stored in the first KB+1 rows of * the array. * * LDBB (input) INTEGER * The leading dimension of the array BB. LDBB >= KB+1. * * X (output) DOUBLE PRECISION array, dimension (LDX,N) * If VECT = 'V', the n-by-n matrix X. * If VECT = 'N', the array X is not referenced. * * LDX (input) INTEGER * The leading dimension of the array X. * LDX >= max(1,N) if VECT = 'V'; LDX >= 1 otherwise. * * WORK (workspace) DOUBLE PRECISION array, dimension (2*N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value. * * ===================================================================== *go to the page top

USAGE: w, z, info, ab, bb = NumRu::Lapack.dsbgv( jobz, uplo, ka, kb, ab, bb, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBGV( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, INFO ) * Purpose * ======= * * 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. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangles of A and B are stored; * = 'L': Lower triangles of A and B are stored. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * KA (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= 0. * * KB (input) INTEGER * The number of superdiagonals of the matrix B if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KB >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KA+1. * * BB (input/output) 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. * * LDBB (input) INTEGER * The leading dimension of the array BB. LDBB >= KB+1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= N. * * WORK (workspace) DOUBLE PRECISION array, dimension (3*N) * * INFO (output) 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. * * ===================================================================== * * .. Local Scalars .. LOGICAL UPPER, WANTZ CHARACTER VECT INTEGER IINFO, INDE, INDWRK * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DPBSTF, DSBGST, DSBTRD, DSTEQR, DSTERF, XERBLA * ..go to the page top

USAGE: w, z, work, iwork, info, ab, bb = NumRu::Lapack.dsbgvd( jobz, uplo, ka, kb, ab, bb, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBGVD( JOBZ, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO ) * Purpose * ======= * * 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. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangles of A and B are stored; * = 'L': Lower triangles of A and B are stored. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * KA (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= 0. * * KB (input) INTEGER * The number of superdiagonals of the matrix B if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KB >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KA+1. * * BB (input/output) 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. * * LDBB (input) INTEGER * The leading dimension of the array BB. LDBB >= KB+1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. * If N <= 1, LWORK >= 1. * If JOBZ = 'N' and N > 1, LWORK >= 3*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. * * IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) * On exit, if LIWORK > 0, IWORK(1) returns the optimal LIWORK. * * LIWORK (input) 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. * * INFO (output) 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. * * Further Details * =============== * * Based on contributions by * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== *go to the page top

USAGE: q, m, w, z, work, iwork, ifail, info, ab, bb = NumRu::Lapack.dsbgvx( jobz, range, uplo, ka, kb, ab, bb, vl, vu, il, iu, abstol, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBGVX( 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 ) * Purpose * ======= * * 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. * * Arguments * ========= * * JOBZ (input) CHARACTER*1 * = 'N': Compute eigenvalues only; * = 'V': Compute eigenvalues and eigenvectors. * * RANGE (input) CHARACTER*1 * = 'A': all eigenvalues will be found. * = 'V': all eigenvalues in the half-open interval (VL,VU] * will be found. * = 'I': the IL-th through IU-th eigenvalues will be found. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangles of A and B are stored; * = 'L': Lower triangles of A and B are stored. * * N (input) INTEGER * The order of the matrices A and B. N >= 0. * * KA (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KA >= 0. * * KB (input) INTEGER * The number of superdiagonals of the matrix B if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KB >= 0. * * AB (input/output) 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. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KA+1. * * BB (input/output) 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. * * LDBB (input) INTEGER * The leading dimension of the array BB. LDBB >= KB+1. * * Q (output) 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. * * LDQ (input) INTEGER * The leading dimension of the array Q. If JOBZ = 'N', * LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). * * VL (input) DOUBLE PRECISION * VU (input) DOUBLE PRECISION * If RANGE='V', the lower and upper bounds of the interval to * be searched for eigenvalues. VL < VU. * Not referenced if RANGE = 'A' or 'I'. * * IL (input) INTEGER * IU (input) INTEGER * If RANGE='I', the indices (in ascending order) of the * smallest and largest eigenvalues to be returned. * 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. * Not referenced if RANGE = 'A' or 'V'. * * ABSTOL (input) DOUBLE PRECISION * The absolute error tolerance for the eigenvalues. * An approximate eigenvalue is accepted as converged * when it is determined to lie in an interval [a,b] * of width less than or equal to * * ABSTOL + EPS * max( |a|,|b| ) , * * where EPS is the machine precision. If ABSTOL is less than * or equal to zero, then EPS*|T| will be used in its place, * where |T| is the 1-norm of the tridiagonal matrix obtained * by reducing A to tridiagonal form. * * 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'). * * M (output) INTEGER * The total number of eigenvalues found. 0 <= M <= N. * If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. * * W (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, the eigenvalues in ascending order. * * Z (output) 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. * * LDZ (input) INTEGER * The leading dimension of the array Z. LDZ >= 1, and if * JOBZ = 'V', LDZ >= max(1,N). * * WORK (workspace/output) DOUBLE PRECISION array, dimension (7*N) * * IWORK (workspace/output) INTEGER array, dimension (5*N) * * IFAIL (output) 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. * * INFO (output) 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. * * Further Details * =============== * * Based on contributions by * Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA * * ===================================================================== *go to the page top

USAGE: d, e, info, ab, q = NumRu::Lapack.dsbtrd( vect, uplo, kd, ab, q, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO ) * Purpose * ======= * * DSBTRD reduces a real symmetric band matrix A to symmetric * tridiagonal form T by an orthogonal similarity transformation: * Q**T * A * Q = T. * * Arguments * ========= * * VECT (input) CHARACTER*1 * = 'N': do not form Q; * = 'V': form Q; * = 'U': update a matrix X, by forming X*Q. * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * KD (input) INTEGER * The number of superdiagonals of the matrix A if UPLO = 'U', * or the number of subdiagonals if UPLO = 'L'. KD >= 0. * * AB (input/output) 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, the diagonal elements of AB are overwritten by the * diagonal elements of the tridiagonal matrix T; if KD > 0, the * elements on the first superdiagonal (if UPLO = 'U') or the * first subdiagonal (if UPLO = 'L') are overwritten by the * off-diagonal elements of T; the rest of AB is overwritten by * values generated during the reduction. * * LDAB (input) INTEGER * The leading dimension of the array AB. LDAB >= KD+1. * * D (output) DOUBLE PRECISION array, dimension (N) * The diagonal elements of the tridiagonal matrix T. * * E (output) DOUBLE PRECISION array, dimension (N-1) * The off-diagonal elements of the tridiagonal matrix T: * E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. * * Q (input/output) DOUBLE PRECISION array, dimension (LDQ,N) * On entry, if VECT = 'U', then Q must contain an N-by-N * matrix X; if VECT = 'N' or 'V', then Q need not be set. * * On exit: * if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; * if VECT = 'U', Q contains the product X*Q; * if VECT = 'N', the array Q is not referenced. * * LDQ (input) INTEGER * The leading dimension of the array Q. * LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. * * WORK (workspace) DOUBLE PRECISION array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * Modified by Linda Kaufman, Bell Labs. * * ===================================================================== *go to the page top

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