#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dgeev_(char *jobvl, char *jobvr, integer *n, doublereal * a, integer *lda, doublereal *wr, doublereal *wi, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, doublereal *work, integer *lwork, integer *info) { /* -- LAPACK driver routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= DGEEV computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The left eigenvector u(j) of A satisfies u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j). The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Arguments ========= JOBVL (input) CHARACTER*1 = 'N': left eigenvectors of A are not computed; = 'V': left eigenvectors of A are computed. JOBVR (input) CHARACTER*1 = 'N': right eigenvectors of A are not computed; = 'V': right eigenvectors of A are computed. N (input) INTEGER The order of the matrix A. N >= 0. A (input/output) DOUBLE PRECISION array, dimension (LDA,N) On entry, the N-by-N matrix A. On exit, A has been overwritten. LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). WR (output) DOUBLE PRECISION array, dimension (N) WI (output) DOUBLE PRECISION array, dimension (N) WR and WI contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first. VL (output) DOUBLE PRECISION array, dimension (LDVL,N) If JOBVL = 'V', the left eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. If JOBVL = 'N', VL is not referenced. If the j-th eigenvalue is real, then u(j) = VL(:,j), the j-th column of VL. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1). LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N. VR (output) DOUBLE PRECISION array, dimension (LDVR,N) If JOBVR = 'V', the right eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. If JOBVR = 'N', VR is not referenced. If the j-th eigenvalue is real, then v(j) = VR(:,j), the j-th column of VR. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and v(j+1) = VR(:,j) - i*VR(:,j+1). LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N. WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) INTEGER The dimension of the array WORK. LWORK >= max(1,3*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N. For good performance, LWORK must generally be larger. 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. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value. > 0: if INFO = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements i+1:N of WR and WI contain eigenvalues which have converged. ===================================================================== Test the input arguments Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static integer c__0 = 0; static integer c_n1 = -1; /* System generated locals */ integer a_dim1, a_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2, i__3; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer i__, k; static doublereal r__, cs, sn; static integer ihi; static doublereal scl; static integer ilo; static doublereal dum[1], eps; static integer ibal; static char side[1]; static doublereal anrm; static integer ierr, itau; extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *); static integer iwrk, nout; extern doublereal dnrm2_(integer *, doublereal *, integer *); extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, integer *); extern logical lsame_(char *, char *); extern doublereal dlapy2_(doublereal *, doublereal *); extern /* Subroutine */ int dlabad_(doublereal *, doublereal *), dgebak_( char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *), dgebal_(char *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *); static logical scalea; extern doublereal dlamch_(char *); static doublereal cscale; extern doublereal dlange_(char *, integer *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dgehrd_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); extern integer idamax_(integer *, doublereal *, integer *); extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *), xerbla_(char *, integer *); static logical select[1]; extern integer ilaenv_(integer *, char *, char *, integer *, integer *, integer *, integer *, ftnlen, ftnlen); static doublereal bignum; extern /* Subroutine */ int dorghr_(integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *), dhseqr_(char *, char *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *), dtrevc_(char *, char *, logical *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *); static integer minwrk, maxwrk; static logical wantvl; static doublereal smlnum; static integer hswork; static logical lquery, wantvr; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; --wr; --wi; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --work; /* Function Body */ *info = 0; lquery = *lwork == -1; wantvl = lsame_(jobvl, "V"); wantvr = lsame_(jobvr, "V"); if (! wantvl && ! lsame_(jobvl, "N")) { *info = -1; } else if (! wantvr && ! lsame_(jobvr, "N")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*lda < max(1,*n)) { *info = -5; } else if (*ldvl < 1 || wantvl && *ldvl < *n) { *info = -9; } else if (*ldvr < 1 || wantvr && *ldvr < *n) { *info = -11; } /* Compute workspace (Note: Comments in the code beginning "Workspace:" describe the minimal amount of workspace needed at that point in the code, as well as the preferred amount for good performance. NB refers to the optimal block size for the immediately following subroutine, as returned by ILAENV. HSWORK refers to the workspace preferred by DHSEQR, as calculated below. HSWORK is computed assuming ILO=1 and IHI=N, the worst case.) */ if (*info == 0) { if (*n == 0) { minwrk = 1; maxwrk = 1; } else { maxwrk = (*n << 1) + *n * ilaenv_(&c__1, "DGEHRD", " ", n, &c__1, n, &c__0, (ftnlen)6, (ftnlen)1); if (wantvl) { minwrk = *n << 2; /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1, "DORGHR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); dhseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[ 1], &vl[vl_offset], ldvl, &work[1], &c_n1, info); hswork = (integer) work[1]; /* Computing MAX */ i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = * n + hswork; maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n << 2; maxwrk = max(i__1,i__2); } else if (wantvr) { minwrk = *n << 2; /* Computing MAX */ i__1 = maxwrk, i__2 = (*n << 1) + (*n - 1) * ilaenv_(&c__1, "DORGHR", " ", n, &c__1, n, &c_n1, (ftnlen)6, (ftnlen)1); maxwrk = max(i__1,i__2); dhseqr_("S", "V", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[ 1], &vr[vr_offset], ldvr, &work[1], &c_n1, info); hswork = (integer) work[1]; /* Computing MAX */ i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = * n + hswork; maxwrk = max(i__1,i__2); /* Computing MAX */ i__1 = maxwrk, i__2 = *n << 2; maxwrk = max(i__1,i__2); } else { minwrk = *n * 3; dhseqr_("E", "N", n, &c__1, n, &a[a_offset], lda, &wr[1], &wi[ 1], &vr[vr_offset], ldvr, &work[1], &c_n1, info); hswork = (integer) work[1]; /* Computing MAX */ i__1 = maxwrk, i__2 = *n + 1, i__1 = max(i__1,i__2), i__2 = * n + hswork; maxwrk = max(i__1,i__2); } maxwrk = max(maxwrk,minwrk); } work[1] = (doublereal) maxwrk; if (*lwork < minwrk && ! lquery) { *info = -13; } } if (*info != 0) { i__1 = -(*info); xerbla_("DGEEV ", &i__1); return 0; } else if (lquery) { return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S"); bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); smlnum = sqrt(smlnum) / eps; bignum = 1. / smlnum; /* Scale A if max element outside range [SMLNUM,BIGNUM] */ anrm = dlange_("M", n, n, &a[a_offset], lda, dum); scalea = FALSE_; if (anrm > 0. && anrm < smlnum) { scalea = TRUE_; cscale = smlnum; } else if (anrm > bignum) { scalea = TRUE_; cscale = bignum; } if (scalea) { dlascl_("G", &c__0, &c__0, &anrm, &cscale, n, n, &a[a_offset], lda, & ierr); } /* Balance the matrix (Workspace: need N) */ ibal = 1; dgebal_("B", n, &a[a_offset], lda, &ilo, &ihi, &work[ibal], &ierr); /* Reduce to upper Hessenberg form (Workspace: need 3*N, prefer 2*N+N*NB) */ itau = ibal + *n; iwrk = itau + *n; i__1 = *lwork - iwrk + 1; dgehrd_(n, &ilo, &ihi, &a[a_offset], lda, &work[itau], &work[iwrk], &i__1, &ierr); if (wantvl) { /* Want left eigenvectors Copy Householder vectors to VL */ *(unsigned char *)side = 'L'; dlacpy_("L", n, n, &a[a_offset], lda, &vl[vl_offset], ldvl) ; /* Generate orthogonal matrix in VL (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) */ i__1 = *lwork - iwrk + 1; dorghr_(n, &ilo, &ihi, &vl[vl_offset], ldvl, &work[itau], &work[iwrk], &i__1, &ierr); /* Perform QR iteration, accumulating Schur vectors in VL (Workspace: need N+1, prefer N+HSWORK (see comments) ) */ iwrk = itau; i__1 = *lwork - iwrk + 1; dhseqr_("S", "V", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], & vl[vl_offset], ldvl, &work[iwrk], &i__1, info); if (wantvr) { /* Want left and right eigenvectors Copy Schur vectors to VR */ *(unsigned char *)side = 'B'; dlacpy_("F", n, n, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr); } } else if (wantvr) { /* Want right eigenvectors Copy Householder vectors to VR */ *(unsigned char *)side = 'R'; dlacpy_("L", n, n, &a[a_offset], lda, &vr[vr_offset], ldvr) ; /* Generate orthogonal matrix in VR (Workspace: need 3*N-1, prefer 2*N+(N-1)*NB) */ i__1 = *lwork - iwrk + 1; dorghr_(n, &ilo, &ihi, &vr[vr_offset], ldvr, &work[itau], &work[iwrk], &i__1, &ierr); /* Perform QR iteration, accumulating Schur vectors in VR (Workspace: need N+1, prefer N+HSWORK (see comments) ) */ iwrk = itau; i__1 = *lwork - iwrk + 1; dhseqr_("S", "V", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], & vr[vr_offset], ldvr, &work[iwrk], &i__1, info); } else { /* Compute eigenvalues only (Workspace: need N+1, prefer N+HSWORK (see comments) ) */ iwrk = itau; i__1 = *lwork - iwrk + 1; dhseqr_("E", "N", n, &ilo, &ihi, &a[a_offset], lda, &wr[1], &wi[1], & vr[vr_offset], ldvr, &work[iwrk], &i__1, info); } /* If INFO > 0 from DHSEQR, then quit */ if (*info > 0) { goto L50; } if (wantvl || wantvr) { /* Compute left and/or right eigenvectors (Workspace: need 4*N) */ dtrevc_(side, "B", select, n, &a[a_offset], lda, &vl[vl_offset], ldvl, &vr[vr_offset], ldvr, n, &nout, &work[iwrk], &ierr); } if (wantvl) { /* Undo balancing of left eigenvectors (Workspace: need N) */ dgebak_("B", "L", n, &ilo, &ihi, &work[ibal], n, &vl[vl_offset], ldvl, &ierr); /* Normalize left eigenvectors and make largest component real */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (wi[i__] == 0.) { scl = 1. / dnrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1); dscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1); } else if (wi[i__] > 0.) { d__1 = dnrm2_(n, &vl[i__ * vl_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vl[(i__ + 1) * vl_dim1 + 1], &c__1); scl = 1. / dlapy2_(&d__1, &d__2); dscal_(n, &scl, &vl[i__ * vl_dim1 + 1], &c__1); dscal_(n, &scl, &vl[(i__ + 1) * vl_dim1 + 1], &c__1); i__2 = *n; for (k = 1; k <= i__2; ++k) { /* Computing 2nd power */ d__1 = vl[k + i__ * vl_dim1]; /* Computing 2nd power */ d__2 = vl[k + (i__ + 1) * vl_dim1]; work[iwrk + k - 1] = d__1 * d__1 + d__2 * d__2; /* L10: */ } k = idamax_(n, &work[iwrk], &c__1); dlartg_(&vl[k + i__ * vl_dim1], &vl[k + (i__ + 1) * vl_dim1], &cs, &sn, &r__); drot_(n, &vl[i__ * vl_dim1 + 1], &c__1, &vl[(i__ + 1) * vl_dim1 + 1], &c__1, &cs, &sn); vl[k + (i__ + 1) * vl_dim1] = 0.; } /* L20: */ } } if (wantvr) { /* Undo balancing of right eigenvectors (Workspace: need N) */ dgebak_("B", "R", n, &ilo, &ihi, &work[ibal], n, &vr[vr_offset], ldvr, &ierr); /* Normalize right eigenvectors and make largest component real */ i__1 = *n; for (i__ = 1; i__ <= i__1; ++i__) { if (wi[i__] == 0.) { scl = 1. / dnrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1); dscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1); } else if (wi[i__] > 0.) { d__1 = dnrm2_(n, &vr[i__ * vr_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vr[(i__ + 1) * vr_dim1 + 1], &c__1); scl = 1. / dlapy2_(&d__1, &d__2); dscal_(n, &scl, &vr[i__ * vr_dim1 + 1], &c__1); dscal_(n, &scl, &vr[(i__ + 1) * vr_dim1 + 1], &c__1); i__2 = *n; for (k = 1; k <= i__2; ++k) { /* Computing 2nd power */ d__1 = vr[k + i__ * vr_dim1]; /* Computing 2nd power */ d__2 = vr[k + (i__ + 1) * vr_dim1]; work[iwrk + k - 1] = d__1 * d__1 + d__2 * d__2; /* L30: */ } k = idamax_(n, &work[iwrk], &c__1); dlartg_(&vr[k + i__ * vr_dim1], &vr[k + (i__ + 1) * vr_dim1], &cs, &sn, &r__); drot_(n, &vr[i__ * vr_dim1 + 1], &c__1, &vr[(i__ + 1) * vr_dim1 + 1], &c__1, &cs, &sn); vr[k + (i__ + 1) * vr_dim1] = 0.; } /* L40: */ } } /* Undo scaling if necessary */ L50: if (scalea) { i__1 = *n - *info; /* Computing MAX */ i__3 = *n - *info; i__2 = max(i__3,1); dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[*info + 1], &i__2, &ierr); i__1 = *n - *info; /* Computing MAX */ i__3 = *n - *info; i__2 = max(i__3,1); dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[*info + 1], &i__2, &ierr); if (*info > 0) { i__1 = ilo - 1; dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wr[1], n, &ierr); i__1 = ilo - 1; dlascl_("G", &c__0, &c__0, &cscale, &anrm, &i__1, &c__1, &wi[1], n, &ierr); } } work[1] = (doublereal) maxwrk; return 0; /* End of DGEEV */ } /* dgeev_ */