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# Error Bounds for the Generalized Symmetric Definite Eigenproblem

Three types of problems must be considered. In all cases A and B are real symmetric (or complex Hermitian) and B is positive definite. These decompositions are computed for real symmetric matrices by the driver routines PxSYGVX (see section 3.2.4). These decompositions are computed for complex Hermitian matrices by the driver routines PxHEGVX (see subsection 3.2.4). In each of the following three decompositions, is real and diagonal with diagonal entries , and the columns of Z are linearly independent vectors. The are called eigenvalues and the are eigenvectors.

1. . The eigendecomposition may be written and (or and if A and B are complex). This may also be written .
2. . The eigendecomposition may be written and ( and if A and B are complex). This may also be written .
3. . The eigendecomposition may be written and ( and if A and B are complex). This may also be written .

The approximate error bounds for the computed eigenvalues are
The approximate error bounds for the computed eigenvectors , which bound the acute angles between the computed eigenvectors and true eigenvectors , are
These bounds are computed differently, depending on which of the above three problems are to be solved. The following code fragments show how.

1. First we consider error bounds for problem 1.

             EPSMCH = PSLAMCH( ICTXT, 'E' )
             UNFL = PSLAMCH( ICTXT, 'U' )
       *     Solve the eigenproblem A - lambda B (ITYPE = 1)
             ITYPE = 1
       *     Compute the norms of A and B
             ANORM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK )
             BNORM = PSLANSY( '1', UPLO, N, B, IB, JB, DESCB, WORK )
       *     The eigenvalues are returned in W
       *     The eigenvectors are returned in A
             SUBROUTINE PSSYGVX( ITYPE, 'V', 'A', UPLO, N, A, IA, JA,
            $                    DESCA, B, IB, JB, DESCB, VL, VU, IL, IU,
            $                    UNFL, M, NZ, W, -1.0, Z, IZ, JZ, DESCZ,
            $                    WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
            $                    GAP, INFO )
             IF( INFO.GT.0 ) THEN
                PRINT *,'PSSYGVX did not converge, or B not positive definite'
             ELSE IF( N.GT.0 ) THEN
       *        Get reciprocal condition number RCONDB of Cholesky factor of B
                CALL PSTRCON( '1', UPLO, 'N', N, B, IB, JB, DESCB, RCONDB,
            $                 WORK, LWORK, IWORK, LIWORK, INFO )
                RCONDB = MAX( RCONDB, EPSMCH )
                CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
                DO 10 I = 1, N
                   EERRBD( I ) = ( EPSMCH / RCONDB**2 ) * ( ANORM / BNORM +
            $                      ABS( W( I ) ) )
                   ZERRBD( I ) = ( EPSMCH / RCONDB**3 ) * ( ( ANORM / BNORM )
            $                     / RCONDZ( I ) + ( ABS( W( I ) ) /
            $                     RCONDZ( I ) ) * RCONDB )
       10       CONTINUE
             END IF

   For example, if ,
   then , , and , and the approximate eigenvalues, approximate error bounds, and true errors are shown below.

2. Problem types 2 and 3 have the same error bounds. We illustrate only type 2.

             EPSMCH = PSLAMCH( ICTXT, 'E' )
       *     Solve the eigenproblem A*B - lambda I (ITYPE = 2)
             ITYPE = 2
       *     Compute the norms of A and B
             ANORM = PSLANSY( '1', UPLO, N, A, IA, JA, DESCA, WORK )
             BNORM = PSLANSY( '1', UPLO, N, B, IB, JB, DESCB, WORK )
       *     The eigenvalues are returned in W
       *     The eigenvectors are returned in A
             SUBROUTINE PSSYGVX( ITYPE, 'V', 'A', UPLO, N, A, IA, JA,
            $                    DESCA, B, IB, JB, DESCB, VL, VU, IL, IU,
            $                    UNFL, M, NZ, W, -1.0, Z, IZ, JZ, DESCZ,
            $                    WORK, LWORK, IWORK, LIWORK, IFAIL, ICLUSTR,
            $                    GAP, INFO )
             IF( INFO.GT.0 .AND. INFO.LE.N ) THEN
                PRINT *,'PSSYGVX did not converge'
             ELSE IF( INFO.GT.N ) THEN
                PRINT *,'B not positive definite'
             ELSE IF( N.GT.0 ) THEN
       *        Get reciprocal condition number RCONDB of Cholesky factor of B
                CALL PSTRCON( '1', UPLO, 'N', N, B, IB, JB, DESCB, RCONDB,
            $                 WORK, LWORK, IWORK, LIWORK, INFO )
                RCONDB = MAX( RCONDB, EPSMCH )
                CALL SDISNA( 'Eigenvectors', N, N, W, RCONDZ, INFO )
                DO 10 I = 1, N
                   EERRBD(I) = ( ANORM * BNORM ) * EPSMCH +
            $                  ( EPSMCH / RCONDB**2 ) * ABS( W( I ) )
                   ZERRBD(I) = ( EPSMCH / RCONDB ) * ( ( ANORM * BNORM ) /
            $                    RCONDZ( I ) + 1.0 / RCONDB )
       10       CONTINUE
             END IF

   For the same A and B as above, the approximate eigenvalues, approximate error bounds, and true errors are shown below.

Further Details: Error Bounds for the Generalized Symmetric Definite Eigenproblem

The error analysis of the driver routine PxSYGVX, or PxHEGVX in the complex case (see section 3.2.4), goes as follows. In all cases is the absolute gap between and the nearest other eigenvalue.

1. . The computed eigenvalues can differ from true eigenvalues by at most about

   The angular difference between the computed eigenvector and a true eigenvector is

2. or . The computed eigenvalues can differ from true eigenvalues by at most about

   The angular difference between the computed eigenvector and a true eigenvector is

The code fragments above replace p(n) by 1 and make sure neither `RCONDB` nor `RCONDZ` is so small as to cause overflow when used as divisors in the expressions for error bounds.

These error bounds are large when B is ill-conditioned with respect to inversion ( is large). Often, the eigenvalues and eigenvectors are much better conditioned than indicated here. We mention two ways to get tighter bounds. The first way is effective when the diagonal entries of B differ widely in magnitude:

1. . Let be a diagonal matrix. Then replace B by DBD and A by DAD in the above bounds.
2. or . Let be a diagonal matrix. Then replace B by DBD and A by in the above bounds.

The second way to get tighter bounds does not actually supply guaranteed bounds, but its estimates are often better in practice. It is not guaranteed because it assumes the algorithm is backward stable, which is not necessarily true when B is ill-conditioned. It estimates the chordal distance between a true eigenvalue and a computed eigenvalue :
To interpret this measure, we write and . Then . In other words, if represents the one-dimensional subspace consisting of the line through the origin with slope , and represents the analogous subspace , then is the sine of the acute angle between these subspaces. Thus, is bounded by one and is small when both arguments are large. It applies only to the first problem, :

> Suppose a computed eigenvalue of is the exact eigenvalue of a perturbed problem . Let be the unit eigenvector () for the exact eigenvalue . Then if is small compared with , and if is small compared with , we have
> Thus is a condition number for eigenvalue .

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