Contents


NAME

     ssygvd - compute all the eigenvalues,  and  optionally,  the
     eigenvectors of a real generalized symmetric-definite eigen-
     problem, of the form A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or
     B*A*x=(lambda)*x

SYNOPSIS

     SUBROUTINE SSYGVD(ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
           LWORK, IWORK, LIWORK, INFO)

     CHARACTER * 1 JOBZ, UPLO
     INTEGER ITYPE, N, LDA, LDB, LWORK, LIWORK, INFO
     INTEGER IWORK(*)
     REAL A(LDA,*), B(LDB,*), W(*), WORK(*)

     SUBROUTINE SSYGVD_64(ITYPE, JOBZ, UPLO, N, A, LDA, B, LDB, W, WORK,
           LWORK, IWORK, LIWORK, INFO)

     CHARACTER * 1 JOBZ, UPLO
     INTEGER*8 ITYPE, N, LDA, LDB, LWORK, LIWORK, INFO
     INTEGER*8 IWORK(*)
     REAL A(LDA,*), B(LDB,*), W(*), WORK(*)

  F95 INTERFACE
     SUBROUTINE SYGVD(ITYPE, JOBZ, UPLO, [N], A, [LDA], B, [LDB], W, [WORK],
            [LWORK], [IWORK], [LIWORK], [INFO])

     CHARACTER(LEN=1) :: JOBZ, UPLO
     INTEGER :: ITYPE, N, LDA, LDB, LWORK, LIWORK, INFO
     INTEGER, DIMENSION(:) :: IWORK
     REAL, DIMENSION(:) :: W, WORK
     REAL, DIMENSION(:,:) :: A, B

     SUBROUTINE SYGVD_64(ITYPE, JOBZ, UPLO, [N], A, [LDA], B, [LDB], W,
            [WORK], [LWORK], [IWORK], [LIWORK], [INFO])

     CHARACTER(LEN=1) :: JOBZ, UPLO
     INTEGER(8) :: ITYPE, N, LDA, LDB, LWORK, LIWORK, INFO
     INTEGER(8), DIMENSION(:) :: IWORK
     REAL, DIMENSION(:) :: W, WORK
     REAL, DIMENSION(:,:) :: A, B

  C INTERFACE
     #include <sunperf.h>

     void ssygvd(int itype, char jobz, char uplo,  int  n,  float
               *a,  int  lda,  float  *b,  int ldb, float *w, int
               *info);

     void ssygvd_64(long itype, char jobz,  char  uplo,  long  n,
               float  *a, long lda, float *b, long ldb, float *w,
               long *info);

PURPOSE

     ssygvd computes all the  eigenvalues,  and  optionally,  the
     eigenvectors of a real generalized symmetric-definite eigen-
     problem, 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
     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)
               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)
               = 'N':  Compute eigenvalues only;
               = 'V':  Compute eigenvalues and eigenvectors.

     UPLO (input)
               = 'U':  Upper triangles of A and B are stored;
               = 'L':  Lower triangles of A and B are stored.

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

     A (input/output)
               On entry, the symmetric matrix A.  If UPLO =  'U',
               the leading N-by-N upper triangular part of A con-
               tains the upper triangular part of the  matrix  A.
               If UPLO = 'L', the leading N-by-N lower triangular
               part of A contains the lower  triangular  part  of
               the matrix A.

               On exit, if JOBZ = 'V', then if INFO = 0,  A  con-
               tains the matrix Z of eigenvectors.  The eigenvec-
               tors 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 on exit the upper triangle (if
               UPLO='U')  or  the lower triangle (if UPLO='L') of
               A, including the diagonal, is destroyed.

     LDA (input)
               The leading dimension of  the  array  A.   LDA  >=
               max(1,N).

     B (input/output)
               On entry, the symmetric matrix B.  If UPLO =  'U',
               the leading N-by-N upper triangular part of B con-
               tains the upper triangular part of the  matrix  B.
               If UPLO = 'L', the leading N-by-N lower triangular
               part of B contains the lower  triangular  part  of
               the matrix B.

               On exit, if INFO <= N, the part  of  B  containing
               the matrix is overwritten by the triangular factor
               U or L from the Cholesky factorization B =  U**T*U
               or B = L*L**T.

     LDB (input)
               The leading dimension of  the  array  B.   LDB  >=
               max(1,N).

     W (output)
               If INFO = 0, the eigenvalues in ascending order.

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

     LWORK (input)
               The dimension of the  array  WORK.   If  N  <=  1,
               LWORK  >=  1.   If  JOBZ = 'N' and N > 1, LWORK >=
               2*N+1.  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 optimal size of
               the WORK array, returns this value  as  the  first
               entry  of  the  WORK  array,  and no error message
               related to LWORK is issued by XERBLA.

     IWORK (workspace/output)
               On exit, if INFO = 0, IWORK(1) returns the optimal
               LIWORK.

     LIWORK (input)
               The dimension of the array  IWORK.   If  N  <=  1,
               LIWORK  >= 1.  If JOBZ  = 'N' and 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 optimal size of
               the IWORK array, returns this value as  the  first
               entry  of  the  IWORK  array, and no error message
               related to LIWORK is issued by XERBLA.

     INFO (output)
               = 0:  successful exit
               < 0:  if INFO = -i, the i-th argument had an ille-
               gal value
               > 0:  SPOTRF or SSYEVD returned an error code:
               <= N:  if INFO = i, SSYEVD failed to  converge;  i
               off-diagonal elements of an intermediate tridiago-
               nal 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