SUBROUTINE SPTCON( N, D, E, ANORM, RCOND, WORK, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, N REAL ANORM, RCOND * .. * .. Array Arguments .. REAL D( * ), E( * ), WORK( * ) * .. * * Purpose * ======= * * SPTCON computes the reciprocal of the condition number (in the * 1-norm) of a real symmetric positive definite tridiagonal matrix * using the factorization A = L*D*L**T or A = U**T*D*U computed by * SPTTRF. * * Norm(inv(A)) is computed by a direct method, and the reciprocal of * the condition number is computed as * RCOND = 1 / (ANORM * norm(inv(A))). * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * D (input) REAL array, dimension (N) * The n diagonal elements of the diagonal matrix D from the * factorization of A, as computed by SPTTRF. * * E (input) REAL array, dimension (N-1) * The (n-1) off-diagonal elements of the unit bidiagonal factor * U or L from the factorization of A, as computed by SPTTRF. * * ANORM (input) REAL * The 1-norm of the original matrix A. * * RCOND (output) REAL * The reciprocal of the condition number of the matrix A, * computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is the * 1-norm of inv(A) computed in this routine. * * WORK (workspace) REAL array, dimension (N) * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * The method used is described in Nicholas J. Higham, "Efficient * Algorithms for Computing the Condition Number of a Tridiagonal * Matrix", SIAM J. Sci. Stat. Comput., Vol. 7, No. 1, January 1986. * * ===================================================================== * * .. Parameters .. REAL ONE, ZERO PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 ) * .. * .. Local Scalars .. INTEGER I, IX REAL AINVNM * .. * .. External Functions .. INTEGER ISAMAX EXTERNAL ISAMAX * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS * .. * .. Executable Statements .. * * Test the input arguments. * INFO = 0 IF( N.LT.0 ) THEN INFO = -1 ELSE IF( ANORM.LT.ZERO ) THEN INFO = -4 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SPTCON', -INFO ) RETURN END IF * * Quick return if possible * RCOND = ZERO IF( N.EQ.0 ) THEN RCOND = ONE RETURN ELSE IF( ANORM.EQ.ZERO ) THEN RETURN END IF * * Check that D(1:N) is positive. * DO 10 I = 1, N IF( D( I ).LE.ZERO ) $ RETURN 10 CONTINUE * * Solve M(A) * x = e, where M(A) = (m(i,j)) is given by * * m(i,j) = abs(A(i,j)), i = j, * m(i,j) = -abs(A(i,j)), i .ne. j, * * and e = [ 1, 1, ..., 1 ]'. Note M(A) = M(L)*D*M(L)'. * * Solve M(L) * x = e. * WORK( 1 ) = ONE DO 20 I = 2, N WORK( I ) = ONE + WORK( I-1 )*ABS( E( I-1 ) ) 20 CONTINUE * * Solve D * M(L)' * x = b. * WORK( N ) = WORK( N ) / D( N ) DO 30 I = N - 1, 1, -1 WORK( I ) = WORK( I ) / D( I ) + WORK( I+1 )*ABS( E( I ) ) 30 CONTINUE * * Compute AINVNM = max(x(i)), 1<=i<=n. * IX = ISAMAX( N, WORK, 1 ) AINVNM = ABS( WORK( IX ) ) * * Compute the reciprocal condition number. * IF( AINVNM.NE.ZERO ) $ RCOND = ( ONE / AINVNM ) / ANORM * RETURN * * End of SPTCON * END