# DOUBLE PRECISION routines for symmetric, packed storage matrix

## dspcon

```USAGE:
rcond, info = NumRu::Lapack.dspcon( uplo, ap, ipiv, anorm, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPCON( UPLO, N, AP, IPIV, ANORM, RCOND, WORK, IWORK, INFO )

*  Purpose
*  =======
*
*  DSPCON estimates the reciprocal of the condition number (in the
*  1-norm) of a real symmetric packed matrix A using the factorization
*  A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*
*  An estimate is obtained for norm(inv(A)), and the reciprocal of the
*  condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by DSPTRF, stored as a
*          packed triangular matrix.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by DSPTRF.
*
*  ANORM   (input) DOUBLE PRECISION
*          The 1-norm of the original matrix A.
*
*  RCOND   (output) DOUBLE PRECISION
*          The reciprocal of the condition number of the matrix A,
*          computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
*          estimate of the 1-norm of inv(A) computed in this routine.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (2*N)
*
*  IWORK   (workspace) INTEGER array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

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

```
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## dspev

```USAGE:
w, z, info, ap = NumRu::Lapack.dspev( jobz, uplo, ap, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPEV( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, INFO )

*  Purpose
*  =======
*
*  DSPEV computes all the eigenvalues and, optionally, eigenvectors of a
*  real symmetric matrix A in packed storage.
*

*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, AP is overwritten by values generated during the
*          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*          and first superdiagonal of the tridiagonal matrix T overwrite
*          the corresponding elements of A, and if UPLO = 'L', the
*          diagonal and first subdiagonal of T overwrite the
*          corresponding elements of A.
*
*  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 (3*N)
*
*  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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## dspevd

```USAGE:
w, z, work, iwork, info, ap = NumRu::Lapack.dspevd( jobz, uplo, ap, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )

*  Purpose
*  =======
*
*  DSPEVD computes all the eigenvalues and, optionally, eigenvectors
*  of a real symmetric matrix A in packed storage. If eigenvectors are
*  desired, it uses a divide and conquer algorithm.
*
*  The divide and conquer algorithm makes very mild assumptions about
*  floating point arithmetic. It will work on machines with a guard
*  digit in add/subtract, or on those binary machines without guard
*  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*  Cray-2. It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.
*

*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, AP is overwritten by values generated during the
*          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*          and first superdiagonal of the tridiagonal matrix T overwrite
*          the corresponding elements of A, and if UPLO = 'L', the
*          diagonal and first subdiagonal of T overwrite the
*          corresponding elements of A.
*
*  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 required LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.
*          If N <= 1,               LWORK must be at least 1.
*          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
*          If JOBZ = 'V' and N > 1, LWORK must be at least
*                                                 1 + 6*N + N**2.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the required sizes of the WORK and IWORK
*          arrays, returns these values as the first entries of the WORK
*          and IWORK arrays, and no error message related to LWORK or
*          LIWORK is issued by XERBLA.
*
*  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
*          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*
*  LIWORK  (input) INTEGER
*          The dimension of the array IWORK.
*          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
*          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*
*          If LIWORK = -1, then a workspace query is assumed; the
*          routine only calculates the required sizes of the WORK and
*          IWORK arrays, returns these values as the first entries of
*          the WORK and IWORK arrays, and no error message related to
*          LWORK or LIWORK is issued by XERBLA.
*
*  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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## dspevx

```USAGE:
m, w, z, ifail, info, ap = NumRu::Lapack.dspevx( jobz, range, uplo, ap, vl, vu, il, iu, abstol, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

*  Purpose
*  =======
*
*  DSPEVX computes selected eigenvalues and, optionally, eigenvectors
*  of a real symmetric matrix A in packed storage.  Eigenvalues/vectors
*  can be selected by specifying either a range of values or a range of
*  indices for the desired eigenvalues.
*

*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, AP is overwritten by values generated during the
*          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*          and first superdiagonal of the tridiagonal matrix T overwrite
*          the corresponding elements of A, and if UPLO = 'L', the
*          diagonal and first subdiagonal of T overwrite the
*          corresponding elements of A.
*
*  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 AP to tridiagonal form.
*
*          Eigenvalues will be computed most accurately when ABSTOL is
*          set to twice the underflow threshold 2*DLAMCH('S'), not zero.
*          If this routine returns with INFO>0, indicating that some
*          eigenvectors did not converge, try setting ABSTOL to
*          2*DLAMCH('S').
*
*          See "Computing Small Singular Values of Bidiagonal Matrices
*          with Guaranteed High Relative Accuracy," by Demmel and
*          Kahan, LAPACK Working Note #3.
*
*  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 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 (8*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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## dspgst

```USAGE:
info, ap = NumRu::Lapack.dspgst( itype, uplo, n, ap, bp, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPGST( ITYPE, UPLO, N, AP, BP, INFO )

*  Purpose
*  =======
*
*  DSPGST reduces a real symmetric-definite generalized eigenproblem
*  to standard form, using packed storage.
*
*  If ITYPE = 1, the problem is A*x = lambda*B*x,
*  and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
*
*  If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
*  B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
*
*  B must have been previously factorized as U**T*U or L*L**T by DPPTRF.
*

*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          = 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
*          = 2 or 3: compute U*A*U**T or L**T*A*L.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored and B is factored as
*                  U**T*U;
*          = 'L':  Lower triangle of A is stored and B is factored as
*                  L*L**T.
*
*  N       (input) INTEGER
*          The order of the matrices A and B.  N >= 0.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, if INFO = 0, the transformed matrix, stored in the
*          same format as A.
*
*  BP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The triangular factor from the Cholesky factorization of B,
*          stored in the same format as A, as returned by DPPTRF.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

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

```
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## dspgv

```USAGE:
w, z, info, ap, bp = NumRu::Lapack.dspgv( itype, jobz, uplo, ap, bp, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPGV( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, INFO )

*  Purpose
*  =======
*
*  DSPGV computes all the eigenvalues and, optionally, the eigenvectors
*  of a real generalized symmetric-definite eigenproblem, of the form
*  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.
*  Here A and B are assumed to be symmetric, stored in packed format,
*  and B is also positive definite.
*

*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          Specifies the problem type to be solved:
*          = 1:  A*x = (lambda)*B*x
*          = 2:  A*B*x = (lambda)*x
*          = 3:  B*A*x = (lambda)*x
*
*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension
*                            (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, the contents of AP are destroyed.
*
*  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          B, packed columnwise in a linear array.  The j-th column of B
*          is stored in the array BP as follows:
*          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*
*          On exit, the triangular factor U or L from the Cholesky
*          factorization B = U**T*U or B = L*L**T, in the same storage
*          format as B.
*
*  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.  The eigenvectors are normalized as follows:
*          if ITYPE = 1 or 2, Z**T*B*Z = I;
*          if ITYPE = 3, Z**T*inv(B)*Z = I.
*          If JOBZ = 'N', then Z is not referenced.
*
*  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 (3*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  DPPTRF or DSPEV returned an error code:
*             <= N:  if INFO = i, DSPEV failed to converge;
*                    i off-diagonal elements of an intermediate
*                    tridiagonal form did not converge to zero.
*             > N:   if INFO = n + i, for 1 <= i <= n, then the leading
*                    minor of order i of B is not positive definite.
*                    The factorization of B could not be completed and
*                    no eigenvalues or eigenvectors were computed.
*

*  =====================================================================
*
*     .. Local Scalars ..
LOGICAL            UPPER, WANTZ
CHARACTER          TRANS
INTEGER            J, NEIG
*     ..
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DPPTRF, DSPEV, DSPGST, DTPMV, DTPSV, XERBLA
*     ..

```
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## dspgvd

```USAGE:
w, z, work, iwork, info, ap, bp = NumRu::Lapack.dspgvd( itype, jobz, uplo, ap, bp, [:lwork => lwork, :liwork => liwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPGVD( ITYPE, JOBZ, UPLO, N, AP, BP, W, Z, LDZ, WORK, LWORK, IWORK, LIWORK, INFO )

*  Purpose
*  =======
*
*  DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
*  of a real generalized symmetric-definite eigenproblem, of the form
*  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
*  B are assumed to be symmetric, stored in packed format, and B is also
*  positive definite.
*  If eigenvectors are desired, it uses a divide and conquer algorithm.
*
*  The divide and conquer algorithm makes very mild assumptions about
*  floating point arithmetic. It will work on machines with a guard
*  digit in add/subtract, or on those binary machines without guard
*  digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*  Cray-2. It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.
*

*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          Specifies the problem type to be solved:
*          = 1:  A*x = (lambda)*B*x
*          = 2:  A*B*x = (lambda)*x
*          = 3:  B*A*x = (lambda)*x
*
*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, the contents of AP are destroyed.
*
*  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          B, packed columnwise in a linear array.  The j-th column of B
*          is stored in the array BP as follows:
*          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*
*          On exit, the triangular factor U or L from the Cholesky
*          factorization B = U**T*U or B = L*L**T, in the same storage
*          format as B.
*
*  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.  The eigenvectors are normalized as follows:
*          if ITYPE = 1 or 2, Z**T*B*Z = I;
*          if ITYPE = 3, Z**T*inv(B)*Z = I.
*          If JOBZ = 'N', then Z is not referenced.
*
*  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 required LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK.
*          If N <= 1,               LWORK >= 1.
*          If JOBZ = 'N' and N > 1, LWORK >= 2*N.
*          If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the required sizes of the WORK and IWORK
*          arrays, returns these values as the first entries of the WORK
*          and IWORK arrays, and no error message related to LWORK or
*          LIWORK is issued by XERBLA.
*
*  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
*          On exit, if INFO = 0, IWORK(1) returns the required 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 required sizes of the WORK and
*          IWORK arrays, returns these values as the first entries of
*          the WORK and IWORK arrays, and no error message related to
*          LWORK or LIWORK is issued by XERBLA.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  DPPTRF or DSPEVD returned an error code:
*             <= N:  if INFO = i, DSPEVD failed to converge;
*                    i off-diagonal elements of an intermediate
*                    tridiagonal form did not converge to zero;
*             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*                    minor of order i of B is not positive definite.
*                    The factorization of B could not be completed and
*                    no eigenvalues or eigenvectors were computed.
*

*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
*  =====================================================================
*

```
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## dspgvx

```USAGE:
m, w, z, ifail, info, ap, bp = NumRu::Lapack.dspgvx( itype, jobz, range, uplo, ap, bp, vl, vu, il, iu, abstol, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

*  Purpose
*  =======
*
*  DSPGVX computes selected eigenvalues, and optionally, eigenvectors
*  of a real generalized symmetric-definite eigenproblem, of the form
*  A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
*  and B are assumed to be symmetric, stored in packed storage, and B
*  is also positive definite.  Eigenvalues and eigenvectors can be
*  selected by specifying either a range of values or a range of indices
*  for the desired eigenvalues.
*

*  Arguments
*  =========
*
*  ITYPE   (input) INTEGER
*          Specifies the problem type to be solved:
*          = 1:  A*x = (lambda)*B*x
*          = 2:  A*B*x = (lambda)*x
*          = 3:  B*A*x = (lambda)*x
*
*  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 and B are stored;
*          = 'L':  Lower triangle of A and B are stored.
*
*  N       (input) INTEGER
*          The order of the matrix pencil (A,B).  N >= 0.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, the contents of AP are destroyed.
*
*  BP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          B, packed columnwise in a linear array.  The j-th column of B
*          is stored in the array BP as follows:
*          if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
*          if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
*
*          On exit, the triangular factor U or L from the Cholesky
*          factorization B = U**T*U or B = L*L**T, in the same storage
*          format as B.
*
*  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)
*          On normal exit, the first M elements contain the selected
*          eigenvalues in ascending order.
*
*  Z       (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
*          If JOBZ = 'N', then Z is not referenced.
*          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*          contain the orthonormal eigenvectors of the matrix A
*          corresponding to the selected eigenvalues, with the i-th
*          column of Z holding the eigenvector associated with W(i).
*          The eigenvectors are normalized as follows:
*          if ITYPE = 1 or 2, Z**T*B*Z = I;
*          if ITYPE = 3, Z**T*inv(B)*Z = I.
*
*          If an eigenvector fails to converge, then that column of Z
*          contains the latest approximation to the eigenvector, and the
*          index of the eigenvector is returned in IFAIL.
*          Note: the user must ensure that at least max(1,M) columns are
*          supplied in the array Z; if RANGE = 'V', the exact value of M
*          is not known in advance and an upper bound must be used.
*
*  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 (8*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:  DPPTRF or DSPEVX returned an error code:
*             <= N:  if INFO = i, DSPEVX failed to converge;
*                    i eigenvectors failed to converge.  Their indices
*                    are stored in array IFAIL.
*             > N:   if INFO = N + i, for 1 <= i <= N, then the leading
*                    minor of order i of B is not positive definite.
*                    The factorization of B could not be completed and
*                    no eigenvalues or eigenvectors were computed.
*

*  Further Details
*  ===============
*
*  Based on contributions by
*     Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
*
* =====================================================================
*
*     .. Local Scalars ..
LOGICAL            ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
CHARACTER          TRANS
INTEGER            J
*     ..
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DPPTRF, DSPEVX, DSPGST, DTPMV, DTPSV, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MIN
*     ..

```
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## dsposv

```USAGE:
x, iter, info, a = NumRu::Lapack.dsposv( uplo, a, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPOSV( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, WORK, SWORK, ITER, INFO )

*  Purpose
*  =======
*
*  DSPOSV computes the solution to a real system of linear equations
*     A * X = B,
*  where A is an N-by-N symmetric positive definite matrix and X and B
*  are N-by-NRHS matrices.
*
*  DSPOSV first attempts to factorize the matrix in SINGLE PRECISION
*  and use this factorization within an iterative refinement procedure
*  to produce a solution with DOUBLE PRECISION normwise backward error
*  quality (see below). If the approach fails the method switches to a
*  DOUBLE PRECISION factorization and solve.
*
*  The iterative refinement is not going to be a winning strategy if
*  the ratio SINGLE PRECISION performance over DOUBLE PRECISION
*  performance is too small. A reasonable strategy should take the
*  number of right-hand sides and the size of the matrix into account.
*  This might be done with a call to ILAENV in the future. Up to now, we
*  always try iterative refinement.
*
*  The iterative refinement process is stopped if
*      ITER > ITERMAX
*  or for all the RHS we have:
*      RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
*  where
*      o ITER is the number of the current iteration in the iterative
*        refinement process
*      o RNRM is the infinity-norm of the residual
*      o XNRM is the infinity-norm of the solution
*      o ANRM is the infinity-operator-norm of the matrix A
*      o EPS is the machine epsilon returned by DLAMCH('Epsilon')
*  The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00
*  respectively.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  A       (input/output) DOUBLE PRECISION array,
*          dimension (LDA,N)
*          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
*          N-by-N upper triangular part of A contains the upper
*          triangular part of the matrix A, and the strictly lower
*          triangular part of A is not referenced.  If UPLO = 'L', the
*          leading N-by-N lower triangular part of A contains the lower
*          triangular part of the matrix A, and the strictly upper
*          triangular part of A is not referenced.
*          On exit, if iterative refinement has been successfully used
*          (INFO.EQ.0 and ITER.GE.0, see description below), then A is
*          unchanged, if double precision factorization has been used
*          (INFO.EQ.0 and ITER.LT.0, see description below), then the
*          array A contains the factor U or L from the Cholesky
*          factorization A = U**T*U or A = L*L**T.
*
*
*  LDA     (input) INTEGER
*          The leading dimension of the array A.  LDA >= max(1,N).
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          The N-by-NRHS right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
*          If INFO = 0, the N-by-NRHS solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (N,NRHS)
*          This array is used to hold the residual vectors.
*
*  SWORK   (workspace) REAL array, dimension (N*(N+NRHS))
*          This array is used to use the single precision matrix and the
*          right-hand sides or solutions in single precision.
*
*  ITER    (output) INTEGER
*          < 0: iterative refinement has failed, double precision
*               factorization has been performed
*               -1 : the routine fell back to full precision for
*                    implementation- or machine-specific reasons
*               -2 : narrowing the precision induced an overflow,
*                    the routine fell back to full precision
*               -3 : failure of SPOTRF
*               -31: stop the iterative refinement after the 30th
*                    iterations
*          > 0: iterative refinement has been sucessfully used.
*               Returns the number of iterations
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, the leading minor of order i of (DOUBLE
*                PRECISION) A is not positive definite, so the
*                factorization could not be completed, and the solution
*                has not been computed.
*
*  =========
*

```
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## dsprfs

```USAGE:
ferr, berr, info, x = NumRu::Lapack.dsprfs( uplo, ap, afp, ipiv, b, x, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPRFS( UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )

*  Purpose
*  =======
*
*  DSPRFS improves the computed solution to a system of linear
*  equations when the coefficient matrix is symmetric indefinite
*  and packed, and provides error bounds and backward error estimates
*  for the solution.
*

*  Arguments
*  =========
*
*  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.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The upper or lower triangle of the symmetric matrix A, packed
*          columnwise in a linear array.  The j-th column of A is stored
*          in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*
*  AFP     (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The factored form of the matrix A.  AFP contains the block
*          diagonal matrix D and the multipliers used to obtain the
*          factor U or L from the factorization A = U*D*U**T or
*          A = L*D*L**T as computed by DSPTRF, stored as a packed
*          triangular matrix.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by DSPTRF.
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          The right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS)
*          On entry, the solution matrix X, as computed by DSPTRS.
*          On exit, the improved solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
*          The estimated forward error bound for each solution vector
*          X(j) (the j-th column of the solution matrix X).
*          If XTRUE is the true solution corresponding to X(j), FERR(j)
*          is an estimated upper bound for the magnitude of the largest
*          element in (X(j) - XTRUE) divided by the magnitude of the
*          largest element in X(j).  The estimate is as reliable as
*          the estimate for RCOND, and is almost always a slight
*          overestimate of the true error.
*
*  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
*          The componentwise relative backward error of each solution
*          vector X(j) (i.e., the smallest relative change in
*          any element of A or B that makes X(j) an exact solution).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
*
*  IWORK   (workspace) INTEGER array, dimension (N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*
*  Internal Parameters
*  ===================
*
*  ITMAX is the maximum number of steps of iterative refinement.
*

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

```
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## dspsv

```USAGE:
ipiv, info, ap, b = NumRu::Lapack.dspsv( uplo, ap, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPSV( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

*  Purpose
*  =======
*
*  DSPSV computes the solution to a real system of linear equations
*     A * X = B,
*  where A is an N-by-N symmetric matrix stored in packed format and X
*  and B are N-by-NRHS matrices.
*
*  The diagonal pivoting method is used to factor A as
*     A = U * D * U**T,  if UPLO = 'U', or
*     A = L * D * L**T,  if UPLO = 'L',
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, D is symmetric and block diagonal with 1-by-1
*  and 2-by-2 diagonal blocks.  The factored form of A is then used to
*  solve the system of equations A * X = B.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*          See below for further details.
*
*          On exit, the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L from the factorization
*          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*          a packed triangular matrix in the same storage format as A.
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D, as
*          determined by DSPTRF.  If IPIV(k) > 0, then rows and columns
*          k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
*          diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
*          then rows and columns k-1 and -IPIV(k) were interchanged and
*          D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
*          IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
*          -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
*          diagonal block.
*
*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          On entry, the N-by-NRHS right hand side matrix B.
*          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
*                has been completed, but the block diagonal matrix D is
*                exactly singular, so the solution could not be
*                computed.
*

*  Further Details
*  ===============
*
*  The packed storage scheme is illustrated by the following example
*  when N = 4, UPLO = 'U':
*
*  Two-dimensional storage of the symmetric matrix A:
*
*     a11 a12 a13 a14
*         a22 a23 a24
*             a33 a34     (aij = aji)
*                 a44
*
*  Packed storage of the upper triangle of A:
*
*  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*
*  =====================================================================
*
*     .. External Functions ..
LOGICAL            LSAME
EXTERNAL           LSAME
*     ..
*     .. External Subroutines ..
EXTERNAL           DSPTRF, DSPTRS, XERBLA
*     ..
*     .. Intrinsic Functions ..
INTRINSIC          MAX
*     ..

```
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## dspsvx

```USAGE:
x, rcond, ferr, berr, info, afp, ipiv = NumRu::Lapack.dspsvx( fact, uplo, ap, afp, ipiv, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPSVX( FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )

*  Purpose
*  =======
*
*  DSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
*  A = L*D*L**T to compute the solution to a real system of linear
*  equations A * X = B, where A is an N-by-N symmetric matrix stored
*  in packed format and X and B are N-by-NRHS matrices.
*
*  Error bounds on the solution and a condition estimate are also
*  provided.
*
*  Description
*  ===========
*
*  The following steps are performed:
*
*  1. If FACT = 'N', the diagonal pivoting method is used to factor A as
*        A = U * D * U**T,  if UPLO = 'U', or
*        A = L * D * L**T,  if UPLO = 'L',
*     where U (or L) is a product of permutation and unit upper (lower)
*     triangular matrices and D is symmetric and block diagonal with
*     1-by-1 and 2-by-2 diagonal blocks.
*
*  2. If some D(i,i)=0, so that D is exactly singular, then the routine
*     returns with INFO = i. Otherwise, the factored form of A is used
*     to estimate the condition number of the matrix A.  If the
*     reciprocal of the condition number is less than machine precision,
*     INFO = N+1 is returned as a warning, but the routine still goes on
*     to solve for X and compute error bounds as described below.
*
*  3. The system of equations is solved for X using the factored form
*     of A.
*
*  4. Iterative refinement is applied to improve the computed solution
*     matrix and calculate error bounds and backward error estimates
*     for it.
*

*  Arguments
*  =========
*
*  FACT    (input) CHARACTER*1
*          Specifies whether or not the factored form of A has been
*          supplied on entry.
*          = 'F':  On entry, AFP and IPIV contain the factored form of
*                  A.  AP, AFP and IPIV will not be modified.
*          = 'N':  The matrix A will be copied to AFP and factored.
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  Upper triangle of A is stored;
*          = 'L':  Lower triangle of A is stored.
*
*  N       (input) INTEGER
*          The number of linear equations, i.e., the order of the
*          matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrices B and X.  NRHS >= 0.
*
*  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The upper or lower triangle of the symmetric matrix A, packed
*          columnwise in a linear array.  The j-th column of A is stored
*          in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*          See below for further details.
*
*  AFP     (input or output) DOUBLE PRECISION array, dimension
*                            (N*(N+1)/2)
*          If FACT = 'F', then AFP is an input argument and on entry
*          contains the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L from the factorization
*          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*          a packed triangular matrix in the same storage format as A.
*
*          If FACT = 'N', then AFP is an output argument and on exit
*          contains the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L from the factorization
*          A = U*D*U**T or A = L*D*L**T as computed by DSPTRF, stored as
*          a packed triangular matrix in the same storage format as A.
*
*  IPIV    (input or output) INTEGER array, dimension (N)
*          If FACT = 'F', then IPIV is an input argument and on entry
*          contains details of the interchanges and the block structure
*          of D, as determined by DSPTRF.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*          If FACT = 'N', then IPIV is an output argument and on exit
*          contains details of the interchanges and the block structure
*          of D, as determined by DSPTRF.
*
*  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          The N-by-NRHS right hand side matrix B.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  X       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
*          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*
*  LDX     (input) INTEGER
*          The leading dimension of the array X.  LDX >= max(1,N).
*
*  RCOND   (output) DOUBLE PRECISION
*          The estimate of the reciprocal condition number of the matrix
*          A.  If RCOND is less than the machine precision (in
*          particular, if RCOND = 0), the matrix is singular to working
*          precision.  This condition is indicated by a return code of
*          INFO > 0.
*
*  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
*          The estimated forward error bound for each solution vector
*          X(j) (the j-th column of the solution matrix X).
*          If XTRUE is the true solution corresponding to X(j), FERR(j)
*          is an estimated upper bound for the magnitude of the largest
*          element in (X(j) - XTRUE) divided by the magnitude of the
*          largest element in X(j).  The estimate is as reliable as
*          the estimate for RCOND, and is almost always a slight
*          overestimate of the true error.
*
*  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
*          The componentwise relative backward error of each solution
*          vector X(j) (i.e., the smallest relative change in
*          any element of A or B that makes X(j) an exact solution).
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (3*N)
*
*  IWORK   (workspace) INTEGER array, dimension (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:  D(i,i) is exactly zero.  The factorization
*                       has been completed but the factor D is exactly
*                       singular, so the solution and error bounds could
*                       not be computed. RCOND = 0 is returned.
*                = N+1: D is nonsingular, but RCOND is less than machine
*                       precision, meaning that the matrix is singular
*                       to working precision.  Nevertheless, the
*                       solution and error bounds are computed because
*                       there are a number of situations where the
*                       computed solution can be more accurate than the
*                       value of RCOND would suggest.
*

*  Further Details
*  ===============
*
*  The packed storage scheme is illustrated by the following example
*  when N = 4, UPLO = 'U':
*
*  Two-dimensional storage of the symmetric matrix A:
*
*     a11 a12 a13 a14
*         a22 a23 a24
*             a33 a34     (aij = aji)
*                 a44
*
*  Packed storage of the upper triangle of A:
*
*  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]
*
*  =====================================================================
*

```
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## dsptrd

```USAGE:
d, e, tau, info, ap = NumRu::Lapack.dsptrd( uplo, ap, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )

*  Purpose
*  =======
*
*  DSPTRD reduces a real symmetric matrix A stored in packed form to
*  symmetric tridiagonal form T by an orthogonal similarity
*  transformation: Q**T * A * Q = T.
*

*  Arguments
*  =========
*
*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*          On exit, if UPLO = 'U', the diagonal and first superdiagonal
*          of A are overwritten by the corresponding elements of the
*          tridiagonal matrix T, and the elements above the first
*          superdiagonal, with the array TAU, represent the orthogonal
*          matrix Q as a product of elementary reflectors; if UPLO
*          = 'L', the diagonal and first subdiagonal of A are over-
*          written by the corresponding elements of the tridiagonal
*          matrix T, and the elements below the first subdiagonal, with
*          the array TAU, represent the orthogonal matrix Q as a product
*          of elementary reflectors. See Further Details.
*
*  D       (output) DOUBLE PRECISION array, dimension (N)
*          The diagonal elements of the tridiagonal matrix T:
*          D(i) = A(i,i).
*
*  E       (output) DOUBLE PRECISION array, dimension (N-1)
*          The off-diagonal elements of the tridiagonal matrix T:
*          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*
*  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
*          The scalar factors of the elementary reflectors (see Further
*          Details).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*

*  Further Details
*  ===============
*
*  If UPLO = 'U', the matrix Q is represented as a product of elementary
*  reflectors
*
*     Q = H(n-1) . . . H(2) H(1).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real scalar, and v is a real vector with
*  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
*  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
*
*  If UPLO = 'L', the matrix Q is represented as a product of elementary
*  reflectors
*
*     Q = H(1) H(2) . . . H(n-1).
*
*  Each H(i) has the form
*
*     H(i) = I - tau * v * v'
*
*  where tau is a real scalar, and v is a real vector with
*  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
*  overwriting A(i+2:n,i), and tau is stored in TAU(i).
*
*  =====================================================================
*

```
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## dsptrf

```USAGE:
ipiv, info, ap = NumRu::Lapack.dsptrf( uplo, ap, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPTRF( UPLO, N, AP, IPIV, INFO )

*  Purpose
*  =======
*
*  DSPTRF computes the factorization of a real symmetric matrix A stored
*  in packed format using the Bunch-Kaufman diagonal pivoting method:
*
*     A = U*D*U**T  or  A = L*D*L**T
*
*  where U (or L) is a product of permutation and unit upper (lower)
*  triangular matrices, and D is symmetric and block diagonal with
*  1-by-1 and 2-by-2 diagonal blocks.
*

*  Arguments
*  =========
*
*  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.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the upper or lower triangle of the symmetric matrix
*          A, packed columnwise in a linear array.  The j-th column of A
*          is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
*
*          On exit, the block diagonal matrix D and the multipliers used
*          to obtain the factor U or L, stored as a packed triangular
*          matrix overwriting A (see below for further details).
*
*  IPIV    (output) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D.
*          If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*          interchanged and D(k,k) is a 1-by-1 diagonal block.
*          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
*          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*
*  INFO    (output) INTEGER
*          = 0: successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*          > 0: if INFO = i, D(i,i) is exactly zero.  The factorization
*               has been completed, but the block diagonal matrix D is
*               exactly singular, and division by zero will occur if it
*               is used to solve a system of equations.
*

*  Further Details
*  ===============
*
*  5-96 - Based on modifications by J. Lewis, Boeing Computer Services
*         Company
*
*  If UPLO = 'U', then A = U*D*U', where
*     U = P(n)*U(n)* ... *P(k)U(k)* ...,
*  i.e., U is a product of terms P(k)*U(k), where k decreases from n to
*  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    v    0   )   k-s
*     U(k) =  (   0    I    0   )   s
*             (   0    0    I   )   n-k
*                k-s   s   n-k
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
*  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
*  and A(k,k), and v overwrites A(1:k-2,k-1:k).
*
*  If UPLO = 'L', then A = L*D*L', where
*     L = P(1)*L(1)* ... *P(k)*L(k)* ...,
*  i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
*  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
*  and 2-by-2 diagonal blocks D(k).  P(k) is a permutation matrix as
*  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
*  that if the diagonal block D(k) is of order s (s = 1 or 2), then
*
*             (   I    0     0   )  k-1
*     L(k) =  (   0    I     0   )  s
*             (   0    v     I   )  n-k-s+1
*                k-1   s  n-k-s+1
*
*  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
*  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
*  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
*
*  =====================================================================
*

```
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## dsptri

```USAGE:
info, ap = NumRu::Lapack.dsptri( uplo, ap, ipiv, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPTRI( UPLO, N, AP, IPIV, WORK, INFO )

*  Purpose
*  =======
*
*  DSPTRI computes the inverse of a real symmetric indefinite matrix
*  A in packed storage using the factorization A = U*D*U**T or
*  A = L*D*L**T computed by DSPTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  AP      (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          On entry, the block diagonal matrix D and the multipliers
*          used to obtain the factor U or L as computed by DSPTRF,
*          stored as a packed triangular matrix.
*
*          On exit, if INFO = 0, the (symmetric) inverse of the original
*          matrix, stored as a packed triangular matrix. The j-th column
*          of inv(A) is stored in the array AP as follows:
*          if UPLO = 'U', AP(i + (j-1)*j/2) = inv(A)(i,j) for 1<=i<=j;
*          if UPLO = 'L',
*             AP(i + (j-1)*(2n-j)/2) = inv(A)(i,j) for j<=i<=n.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by DSPTRF.
*
*  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
*          > 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
*               inverse could not be computed.
*

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

```
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## dsptrs

```USAGE:
info, b = NumRu::Lapack.dsptrs( uplo, ap, ipiv, b, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE DSPTRS( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO )

*  Purpose
*  =======
*
*  DSPTRS solves a system of linear equations A*X = B with a real
*  symmetric matrix A stored in packed format using the factorization
*  A = U*D*U**T or A = L*D*L**T computed by DSPTRF.
*

*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          Specifies whether the details of the factorization are stored
*          as an upper or lower triangular matrix.
*          = 'U':  Upper triangular, form is A = U*D*U**T;
*          = 'L':  Lower triangular, form is A = L*D*L**T.
*
*  N       (input) INTEGER
*          The order of the matrix A.  N >= 0.
*
*  NRHS    (input) INTEGER
*          The number of right hand sides, i.e., the number of columns
*          of the matrix B.  NRHS >= 0.
*
*  AP      (input) DOUBLE PRECISION array, dimension (N*(N+1)/2)
*          The block diagonal matrix D and the multipliers used to
*          obtain the factor U or L as computed by DSPTRF, stored as a
*          packed triangular matrix.
*
*  IPIV    (input) INTEGER array, dimension (N)
*          Details of the interchanges and the block structure of D
*          as determined by DSPTRF.
*
*  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
*          On entry, the right hand side matrix B.
*          On exit, the solution matrix X.
*
*  LDB     (input) INTEGER
*          The leading dimension of the array B.  LDB >= max(1,N).
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0: if INFO = -i, the i-th argument had an illegal value
*

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

```
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