c----------------------------------------------------------------------- c\BeginDoc c c\Name: dneigh c c\Description: c Compute the eigenvalues of the current upper Hessenberg matrix c and the corresponding Ritz estimates given the current residual norm. c c\Usage: c call dneigh c ( RNORM, N, H, LDH, RITZR, RITZI, BOUNDS, Q, LDQ, WORKL, IERR ) c c\Arguments c RNORM Double precision scalar. (INPUT) c Residual norm corresponding to the current upper Hessenberg c matrix H. c c N Integer. (INPUT) c Size of the matrix H. c c H Double precision N by N array. (INPUT) c H contains the current upper Hessenberg matrix. c c LDH Integer. (INPUT) c Leading dimension of H exactly as declared in the calling c program. c c RITZR, Double precision arrays of length N. (OUTPUT) c RITZI On output, RITZR(1:N) (resp. RITZI(1:N)) contains the real c (respectively imaginary) parts of the eigenvalues of H. c c BOUNDS Double precision array of length N. (OUTPUT) c On output, BOUNDS contains the Ritz estimates associated with c the eigenvalues RITZR and RITZI. This is equal to RNORM c times the last components of the eigenvectors corresponding c to the eigenvalues in RITZR and RITZI. c c Q Double precision N by N array. (WORKSPACE) c Workspace needed to store the eigenvectors of H. c c LDQ Integer. (INPUT) c Leading dimension of Q exactly as declared in the calling c program. c c WORKL Double precision work array of length N**2 + 3*N. (WORKSPACE) c Private (replicated) array on each PE or array allocated on c the front end. This is needed to keep the full Schur form c of H and also in the calculation of the eigenvectors of H. c c IERR Integer. (OUTPUT) c Error exit flag from dlaqrb or dtrevc. c c\EndDoc c c----------------------------------------------------------------------- c c\BeginLib c c\Local variables: c xxxxxx real c c\Routines called: c dlaqrb ARPACK routine to compute the real Schur form of an c upper Hessenberg matrix and last row of the Schur vectors. c arsecond ARPACK utility routine for timing. c dmout ARPACK utility routine that prints matrices c dvout ARPACK utility routine that prints vectors. c dlacpy LAPACK matrix copy routine. c dlapy2 LAPACK routine to compute sqrt(x**2+y**2) carefully. c dtrevc LAPACK routine to compute the eigenvectors of a matrix c in upper quasi-triangular form c dgemv Level 2 BLAS routine for matrix vector multiplication. c dcopy Level 1 BLAS that copies one vector to another . c dnrm2 Level 1 BLAS that computes the norm of a vector. c dscal Level 1 BLAS that scales a vector. c c c\Author c Danny Sorensen Phuong Vu c Richard Lehoucq CRPC / Rice University c Dept. of Computational & Houston, Texas c Applied Mathematics c Rice University c Houston, Texas c c\Revision history: c xx/xx/92: Version ' 2.1' c c\SCCS Information: @(#) c FILE: neigh.F SID: 2.3 DATE OF SID: 4/20/96 RELEASE: 2 c c\Remarks c None c c\EndLib c c----------------------------------------------------------------------- c subroutine dneigh (rnorm, n, h, ldh, ritzr, ritzi, bounds, & q, ldq, workl, ierr) c c %----------------------------------------------------% c | Include files for debugging and timing information | c %----------------------------------------------------% c include 'debug.h' include 'stat.h' c c %------------------% c | Scalar Arguments | c %------------------% c integer ierr, n, ldh, ldq Double precision & rnorm c c %-----------------% c | Array Arguments | c %-----------------% c Double precision & bounds(n), h(ldh,n), q(ldq,n), ritzi(n), ritzr(n), & workl(n*(n+3)) c c %------------% c | Parameters | c %------------% c Double precision & one, zero parameter (one = 1.0D+0, zero = 0.0D+0) c c %------------------------% c | Local Scalars & Arrays | c %------------------------% c logical select(1) integer i, iconj, msglvl Double precision & temp, vl(1) c c %----------------------% c | External Subroutines | c %----------------------% c external dcopy, dlacpy, dlaqrb, dtrevc, dvout c c %--------------------% c | External Functions | c %--------------------% c Double precision & dlapy2, dnrm2 external dlapy2, dnrm2 c c %---------------------% c | Intrinsic Functions | c %---------------------% c intrinsic abs c c %-----------------------% c | Executable Statements | c %-----------------------% c c c %-------------------------------% c | Initialize timing statistics | c | & message level for debugging | c %-------------------------------% c call arsecond (t0) msglvl = mneigh c if (msglvl .gt. 2) then call dmout (logfil, n, n, h, ldh, ndigit, & '_neigh: Entering upper Hessenberg matrix H ') end if c c %-----------------------------------------------------------% c | 1. Compute the eigenvalues, the last components of the | c | corresponding Schur vectors and the full Schur form T | c | of the current upper Hessenberg matrix H. | c | dlaqrb returns the full Schur form of H in WORKL(1:N**2) | c | and the last components of the Schur vectors in BOUNDS. | c %-----------------------------------------------------------% c call dlacpy ('All', n, n, h, ldh, workl, n) call dlaqrb (.true., n, 1, n, workl, n, ritzr, ritzi, bounds, & ierr) if (ierr .ne. 0) go to 9000 c if (msglvl .gt. 1) then call dvout (logfil, n, bounds, ndigit, & '_neigh: last row of the Schur matrix for H') end if c c %-----------------------------------------------------------% c | 2. Compute the eigenvectors of the full Schur form T and | c | apply the last components of the Schur vectors to get | c | the last components of the corresponding eigenvectors. | c | Remember that if the i-th and (i+1)-st eigenvalues are | c | complex conjugate pairs, then the real & imaginary part | c | of the eigenvector components are split across adjacent | c | columns of Q. | c %-----------------------------------------------------------% c call dtrevc ('R', 'A', select, n, workl, n, vl, n, q, ldq, & n, n, workl(n*n+1), ierr) c if (ierr .ne. 0) go to 9000 c c %------------------------------------------------% c | Scale the returning eigenvectors so that their | c | euclidean norms are all one. LAPACK subroutine | c | dtrevc returns each eigenvector normalized so | c | that the element of largest magnitude has | c | magnitude 1; here the magnitude of a complex | c | number (x,y) is taken to be |x| + |y|. | c %------------------------------------------------% c iconj = 0 do 10 i=1, n if ( abs( ritzi(i) ) .le. zero ) then c c %----------------------% c | Real eigenvalue case | c %----------------------% c temp = dnrm2( n, q(1,i), 1 ) call dscal ( n, one / temp, q(1,i), 1 ) else c c %-------------------------------------------% c | Complex conjugate pair case. Note that | c | since the real and imaginary part of | c | the eigenvector are stored in consecutive | c | columns, we further normalize by the | c | square root of two. | c %-------------------------------------------% c if (iconj .eq. 0) then temp = dlapy2( dnrm2( n, q(1,i), 1 ), & dnrm2( n, q(1,i+1), 1 ) ) call dscal ( n, one / temp, q(1,i), 1 ) call dscal ( n, one / temp, q(1,i+1), 1 ) iconj = 1 else iconj = 0 end if end if 10 continue c call dgemv ('T', n, n, one, q, ldq, bounds, 1, zero, workl, 1) c if (msglvl .gt. 1) then call dvout (logfil, n, workl, ndigit, & '_neigh: Last row of the eigenvector matrix for H') end if c c %----------------------------% c | Compute the Ritz estimates | c %----------------------------% c iconj = 0 do 20 i = 1, n if ( abs( ritzi(i) ) .le. zero ) then c c %----------------------% c | Real eigenvalue case | c %----------------------% c bounds(i) = rnorm * abs( workl(i) ) else c c %-------------------------------------------% c | Complex conjugate pair case. Note that | c | since the real and imaginary part of | c | the eigenvector are stored in consecutive | c | columns, we need to take the magnitude | c | of the last components of the two vectors | c %-------------------------------------------% c if (iconj .eq. 0) then bounds(i) = rnorm * dlapy2( workl(i), workl(i+1) ) bounds(i+1) = bounds(i) iconj = 1 else iconj = 0 end if end if 20 continue c if (msglvl .gt. 2) then call dvout (logfil, n, ritzr, ndigit, & '_neigh: Real part of the eigenvalues of H') call dvout (logfil, n, ritzi, ndigit, & '_neigh: Imaginary part of the eigenvalues of H') call dvout (logfil, n, bounds, ndigit, & '_neigh: Ritz estimates for the eigenvalues of H') end if c call arsecond (t1) tneigh = tneigh + (t1 - t0) c 9000 continue return c c %---------------% c | End of dneigh | c %---------------% c end