SPEVD - 5.2 English - 68552

SPEVD computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices.

Purpose:

    SPEVD 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.

Parameters:

• JOBZ – [in]

  JOBZ is CHARACTER*1

  = ‘N’: Compute eigenvalues only;

  = ‘V’: Compute eigenvalues and eigenvectors.

• UPLO – [in]

  UPLO is CHARACTER*1

  = ‘U’: Upper triangle of A is stored;

  = ‘L’: Lower triangle of A is stored.

• N – [in]

  N is INTEGER

  The order of the matrix A. N >= 0.

• AP – [inout]

  AP is REAL 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 – [out]

  W is REAL array, dimension (N)

  If INFO = 0, the eigenvalues in ascending order.

• Z – [out]

  Z is REAL 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 – [in]

  LDZ is INTEGER

  The leading dimension of the array Z. LDZ >= 1, and if JOBZ = ‘V’, LDZ >= fla_max(1,N).

• WORK – [out]

  WORK is REAL array, dimension (MAX(1,LWORK))

  On exit, if INFO = 0, WORK(1) returns the required LWORK.

• LWORK – [in]

  LWORK is 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 – [out]

  IWORK is INTEGER array, dimension (MAX(1,LIWORK))

  On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

• LIWORK – [in]

  LIWORK is 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 – [out]

  INFO is 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.