SPGV - 5.2 English - 68552

SPGV computes all the eigenvalues.

Purpose:

    SPGV 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.

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 (3*N)

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

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

  > 0: SPPTRF or SSPEV returned an error code:

  <= N: if INFO = i, SSPEV 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.