SSPGVD 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.

ITYPE

          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

          JOBZ is CHARACTER*1


          = 'N':  Compute eigenvalues only;


          = 'V':  Compute eigenvalues and eigenvectors.

UPLO

          UPLO is CHARACTER*1


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


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

N

          N is INTEGER


          The order of the matrices A and B.  N >= 0.

AP

          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

          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

          W is REAL array, dimension (N)


          If INFO = 0, the eigenvalues in ascending order.

Z

          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

          LDZ is INTEGER


          The leading dimension of the array Z.  LDZ >= 1, and if


          JOBZ = 'V', LDZ >= max(1,N).

WORK

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


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

LWORK

          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

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


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

LIWORK

          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

          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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA