# DOUBLE PRECISION routines for (real) symmetric band matrix

## dsbev

```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.
*

*  =====================================================================
*

```
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## dsbevd

```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.
*

*  =====================================================================
*

```
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## dsbevx

```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.
*

*  =====================================================================
*

```
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## dsbgst

```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.
*

*  =====================================================================
*

```
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## dsbgv

```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
*     ..

```
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## dsbgvd

```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
*
*  =====================================================================
*

```
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## dsbgvx

```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
*
*  =====================================================================
*

```
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## dsbtrd

```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.
*
*  =====================================================================
*

```
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