SUBROUTINE SLAED1( N, D, Q, LDQ, INDXQ, RHO, CUTPNT, WORK, IWORK, \$ INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER CUTPNT, INFO, LDQ, N REAL RHO * .. * .. Array Arguments .. INTEGER INDXQ( * ), IWORK( * ) REAL D( * ), Q( LDQ, * ), WORK( * ) * .. * * Purpose * ======= * * SLAED1 computes the updated eigensystem of a diagonal * matrix after modification by a rank-one symmetric matrix. This * routine is used only for the eigenproblem which requires all * eigenvalues and eigenvectors of a tridiagonal matrix. SLAED7 handles * the case in which eigenvalues only or eigenvalues and eigenvectors * of a full symmetric matrix (which was reduced to tridiagonal form) * are desired. * * T = Q(in) ( D(in) + RHO * Z*Z' ) Q'(in) = Q(out) * D(out) * Q'(out) * * where Z = Q'u, u is a vector of length N with ones in the * CUTPNT and CUTPNT + 1 th elements and zeros elsewhere. * * The eigenvectors of the original matrix are stored in Q, and the * eigenvalues are in D. The algorithm consists of three stages: * * The first stage consists of deflating the size of the problem * when there are multiple eigenvalues or if there is a zero in * the Z vector. For each such occurence the dimension of the * secular equation problem is reduced by one. This stage is * performed by the routine SLAED2. * * The second stage consists of calculating the updated * eigenvalues. This is done by finding the roots of the secular * equation via the routine SLAED4 (as called by SLAED3). * This routine also calculates the eigenvectors of the current * problem. * * The final stage consists of computing the updated eigenvectors * directly using the updated eigenvalues. The eigenvectors for * the current problem are multiplied with the eigenvectors from * the overall problem. * * Arguments * ========= * * N (input) INTEGER * The dimension of the symmetric tridiagonal matrix. N >= 0. * * D (input/output) REAL array, dimension (N) * On entry, the eigenvalues of the rank-1-perturbed matrix. * On exit, the eigenvalues of the repaired matrix. * * Q (input/output) REAL array, dimension (LDQ,N) * On entry, the eigenvectors of the rank-1-perturbed matrix. * On exit, the eigenvectors of the repaired tridiagonal matrix. * * LDQ (input) INTEGER * The leading dimension of the array Q. LDQ >= max(1,N). * * INDXQ (input/output) INTEGER array, dimension (N) * On entry, the permutation which separately sorts the two * subproblems in D into ascending order. * On exit, the permutation which will reintegrate the * subproblems back into sorted order, * i.e. D( INDXQ( I = 1, N ) ) will be in ascending order. * * RHO (input) REAL * The subdiagonal entry used to create the rank-1 modification. * * CUTPNT (input) INTEGER * The location of the last eigenvalue in the leading sub-matrix. * min(1,N) <= CUTPNT <= N/2. * * WORK (workspace) REAL array, dimension (4*N + N**2) * * IWORK (workspace) INTEGER array, dimension (4*N) * * INFO (output) INTEGER * = 0: successful exit. * < 0: if INFO = -i, the i-th argument had an illegal value. * > 0: if INFO = 1, an eigenvalue did not converge * * Further Details * =============== * * Based on contributions by * Jeff Rutter, Computer Science Division, University of California * at Berkeley, USA * Modified by Francoise Tisseur, University of Tennessee. * * ===================================================================== * * .. Local Scalars .. INTEGER COLTYP, CPP1, I, IDLMDA, INDX, INDXC, INDXP, \$ IQ2, IS, IW, IZ, K, N1, N2 * .. * .. External Subroutines .. EXTERNAL SCOPY, SLAED2, SLAED3, SLAMRG, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters. * INFO = 0 * IF( N.LT.0 ) THEN INFO = -1 ELSE IF( LDQ.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( MIN( 1, N / 2 ).GT.CUTPNT .OR. ( N / 2 ).LT.CUTPNT ) THEN INFO = -7 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SLAED1', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) \$ RETURN * * The following values are integer pointers which indicate * the portion of the workspace * used by a particular array in SLAED2 and SLAED3. * IZ = 1 IDLMDA = IZ + N IW = IDLMDA + N IQ2 = IW + N * INDX = 1 INDXC = INDX + N COLTYP = INDXC + N INDXP = COLTYP + N * * * Form the z-vector which consists of the last row of Q_1 and the * first row of Q_2. * CALL SCOPY( CUTPNT, Q( CUTPNT, 1 ), LDQ, WORK( IZ ), 1 ) CPP1 = CUTPNT + 1 CALL SCOPY( N-CUTPNT, Q( CPP1, CPP1 ), LDQ, WORK( IZ+CUTPNT ), 1 ) * * Deflate eigenvalues. * CALL SLAED2( K, N, CUTPNT, D, Q, LDQ, INDXQ, RHO, WORK( IZ ), \$ WORK( IDLMDA ), WORK( IW ), WORK( IQ2 ), \$ IWORK( INDX ), IWORK( INDXC ), IWORK( INDXP ), \$ IWORK( COLTYP ), INFO ) * IF( INFO.NE.0 ) \$ GO TO 20 * * Solve Secular Equation. * IF( K.NE.0 ) THEN IS = ( IWORK( COLTYP )+IWORK( COLTYP+1 ) )*CUTPNT + \$ ( IWORK( COLTYP+1 )+IWORK( COLTYP+2 ) )*( N-CUTPNT ) + IQ2 CALL SLAED3( K, N, CUTPNT, D, Q, LDQ, RHO, WORK( IDLMDA ), \$ WORK( IQ2 ), IWORK( INDXC ), IWORK( COLTYP ), \$ WORK( IW ), WORK( IS ), INFO ) IF( INFO.NE.0 ) \$ GO TO 20 * * Prepare the INDXQ sorting permutation. * N1 = K N2 = N - K CALL SLAMRG( N1, N2, D, 1, -1, INDXQ ) ELSE DO 10 I = 1, N INDXQ( I ) = I 10 CONTINUE END IF * 20 CONTINUE RETURN * * End of SLAED1 * END