SPGVD - 5.2 English - 68552

SPGVD computes all the eigenvalues.

Purpose:

    SPGVD 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.

Parameters:

• ITYPE – [in]

  ITYPE is 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 – [in]

  JOBZ is CHARACTER*1

  = ‘N’: Compute eigenvalues only;

  = ‘V’: Compute eigenvalues and eigenvectors.

• UPLO – [in]

  UPLO is CHARACTER*1

  = ‘U’: Upper triangles of A and B are stored;

  = ‘L’: Lower triangles of A and B are stored.

• N – [in]

  N is INTEGER

  The order of the matrices A and B. 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, the contents of AP are destroyed.

• BP – [inout]

  BP is REAL 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 – [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 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 – [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 >= 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 – [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 >= 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 – [out]

  INFO is INTEGER

  = 0: successful exit

  < 0: if INFO = -i, the i-th argument had an illegal value

  > 0: SPPTRF or SSPEVD returned an error code:

  <= N: if INFO = i, SSPEVD 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.