/* Subroutine */ int dtrsna_(char *job, char *howmny, logical *select, integer *n, doublereal *t, integer *ldt, doublereal *vl, integer * ldvl, doublereal *vr, integer *ldvr, doublereal *s, doublereal *sep, integer *mm, integer *m, doublereal *work, integer *ldwork, integer * iwork, integer *info) { /* System generated locals */ integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, work_dim1, work_offset, i__1, i__2; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ integer i__, j, k, n2; doublereal cs; integer nn, ks; doublereal sn, mu, eps, est; integer kase; doublereal cond; extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); logical pair; integer ierr; doublereal dumm, prod; integer ifst; doublereal lnrm; integer ilst; doublereal rnrm; extern doublereal dnrm2_(integer *, doublereal *, integer *); doublereal prod1, prod2, scale, delta; extern logical lsame_(char *, char *); integer isave[3]; logical wants; doublereal dummy[1]; extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); extern doublereal dlapy2_(doublereal *, doublereal *); extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); extern doublereal dlamch_(char *); extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *); doublereal bignum; logical wantbh; extern /* Subroutine */ int dlaqtr_(logical *, logical *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *), dtrexc_(char *, integer * , doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *); logical somcon; doublereal smlnum; logical wantsp; /* -- LAPACK computational routine (version 3.4.0) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* November 2011 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and test the input parameters */ /* Parameter adjustments */ --select; t_dim1 = *ldt; t_offset = 1 + t_dim1; t -= t_offset; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1; vr -= vr_offset; --s; --sep; work_dim1 = *ldwork; work_offset = 1 + work_dim1; work -= work_offset; --iwork; /* Function Body */ wantbh = lsame_(job, "B"); wants = lsame_(job, "E") || wantbh; wantsp = lsame_(job, "V") || wantbh; somcon = lsame_(howmny, "S"); *info = 0; if (! wants && ! wantsp) { *info = -1; } else if (! lsame_(howmny, "A") && ! somcon) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*ldt < max(1,*n)) { *info = -6; } else if (*ldvl < 1 || wants && *ldvl < *n) { *info = -8; } else if (*ldvr < 1 || wants && *ldvr < *n) { *info = -10; } else { /* Set M to the number of eigenpairs for which condition numbers */ /* are required, and test MM. */ if (somcon) { *m = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (t[k + 1 + k * t_dim1] == 0.) { if (select[k]) { ++(*m); } } else { pair = TRUE_; if (select[k] || select[k + 1]) { *m += 2; } } } else { if (select[*n]) { ++(*m); } } } /* L10: */ } } else { *m = *n; } if (*mm < *m) { *info = -13; } else if (*ldwork < 1 || wantsp && *ldwork < *n) { *info = -16; } } if (*info != 0) { i__1 = -(*info); xerbla_("DTRSNA", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } if (*n == 1) { if (somcon) { if (! select[1]) { return 0; } } if (wants) { s[1] = 1.; } if (wantsp) { sep[1] = (d__1 = t[t_dim1 + 1], abs(d__1)); } return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= i__1; ++k) { /* Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block. */ if (pair) { pair = FALSE_; goto L60; } else { if (k < *n) { pair = t[k + 1 + k * t_dim1] != 0.; } } /* Determine whether condition numbers are required for the k-th */ /* eigenpair. */ if (somcon) { if (pair) { if (! select[k] && ! select[k + 1]) { goto L60; } } else { if (! select[k]) { goto L60; } } } ++ks; if (wants) { /* Compute the reciprocal condition number of the k-th */ /* eigenvalue. */ if (! pair) { /* Real eigenvalue. */ prod = ddot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); rnrm = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); lnrm = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); s[ks] = abs(prod) / (rnrm * lnrm); } else { /* Complex eigenvalue. */ prod1 = ddot_(n, &vr[ks * vr_dim1 + 1], &c__1, &vl[ks * vl_dim1 + 1], &c__1); prod1 += ddot_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1, &vl[(ks + 1) * vl_dim1 + 1], &c__1); prod2 = ddot_(n, &vl[ks * vl_dim1 + 1], &c__1, &vr[(ks + 1) * vr_dim1 + 1], &c__1); prod2 -= ddot_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1, &vr[ks * vr_dim1 + 1], &c__1); d__1 = dnrm2_(n, &vr[ks * vr_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vr[(ks + 1) * vr_dim1 + 1], &c__1); rnrm = dlapy2_(&d__1, &d__2); d__1 = dnrm2_(n, &vl[ks * vl_dim1 + 1], &c__1); d__2 = dnrm2_(n, &vl[(ks + 1) * vl_dim1 + 1], &c__1); lnrm = dlapy2_(&d__1, &d__2); cond = dlapy2_(&prod1, &prod2) / (rnrm * lnrm); s[ks] = cond; s[ks + 1] = cond; } } if (wantsp) { /* Estimate the reciprocal condition number of the k-th */ /* eigenvector. */ /* Copy the matrix T to the array WORK and swap the diagonal */ /* block beginning at T(k,k) to the (1,1) position. */ dlacpy_("Full", n, n, &t[t_offset], ldt, &work[work_offset], ldwork); ifst = k; ilst = 1; dtrexc_("No Q", n, &work[work_offset], ldwork, dummy, &c__1, & ifst, &ilst, &work[(*n + 1) * work_dim1 + 1], &ierr); if (ierr == 1 || ierr == 2) { /* Could not swap because blocks not well separated */ scale = 1.; est = bignum; } else { /* Reordering successful */ if (work[work_dim1 + 2] == 0.) { /* Form C = T22 - lambda*I in WORK(2:N,2:N). */ i__2 = *n; for (i__ = 2; i__ <= i__2; ++i__) { work[i__ + i__ * work_dim1] -= work[work_dim1 + 1]; /* L20: */ } n2 = 1; nn = *n - 1; } else { /* Triangularize the 2 by 2 block by unitary */ /* transformation U = [ cs i*ss ] */ /* [ i*ss cs ]. */ /* such that the (1,1) position of WORK is complex */ /* eigenvalue lambda with positive imaginary part. (2,2) */ /* position of WORK is the complex eigenvalue lambda */ /* with negative imaginary part. */ mu = sqrt((d__1 = work[(work_dim1 << 1) + 1], abs(d__1))) * sqrt((d__2 = work[work_dim1 + 2], abs(d__2))); delta = dlapy2_(&mu, &work[work_dim1 + 2]); cs = mu / delta; sn = -work[work_dim1 + 2] / delta; /* Form */ /* C**T = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] */ /* [ mu ] */ /* [ .. ] */ /* [ .. ] */ /* [ mu ] */ /* where C**T is transpose of matrix C, */ /* and RWORK is stored starting in the N+1-st column of */ /* WORK. */ i__2 = *n; for (j = 3; j <= i__2; ++j) { work[j * work_dim1 + 2] = cs * work[j * work_dim1 + 2] ; work[j + j * work_dim1] -= work[work_dim1 + 1]; /* L30: */ } work[(work_dim1 << 1) + 2] = 0.; work[(*n + 1) * work_dim1 + 1] = mu * 2.; i__2 = *n - 1; for (i__ = 2; i__ <= i__2; ++i__) { work[i__ + (*n + 1) * work_dim1] = sn * work[(i__ + 1) * work_dim1 + 1]; /* L40: */ } n2 = 2; nn = *n - 1 << 1; } /* Estimate norm(inv(C**T)) */ est = 0.; kase = 0; L50: dlacn2_(&nn, &work[(*n + 2) * work_dim1 + 1], &work[(*n + 4) * work_dim1 + 1], &iwork[1], &est, &kase, isave); if (kase != 0) { if (kase == 1) { if (n2 == 1) { /* Real eigenvalue: solve C**T*x = scale*c. */ i__2 = *n - 1; dlaqtr_(&c_true, &c_true, &i__2, &work[(work_dim1 << 1) + 2], ldwork, dummy, &dumm, &scale, &work[(*n + 4) * work_dim1 + 1], &work[(* n + 6) * work_dim1 + 1], &ierr); } else { /* Complex eigenvalue: solve */ /* C**T*(p+iq) = scale*(c+id) in real arithmetic. */ i__2 = *n - 1; dlaqtr_(&c_true, &c_false, &i__2, &work[( work_dim1 << 1) + 2], ldwork, &work[(*n + 1) * work_dim1 + 1], &mu, &scale, &work[(* n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], &ierr); } } else { if (n2 == 1) { /* Real eigenvalue: solve C*x = scale*c. */ i__2 = *n - 1; dlaqtr_(&c_false, &c_true, &i__2, &work[( work_dim1 << 1) + 2], ldwork, dummy, & dumm, &scale, &work[(*n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], & ierr); } else { /* Complex eigenvalue: solve */ /* C*(p+iq) = scale*(c+id) in real arithmetic. */ i__2 = *n - 1; dlaqtr_(&c_false, &c_false, &i__2, &work[( work_dim1 << 1) + 2], ldwork, &work[(*n + 1) * work_dim1 + 1], &mu, &scale, &work[(* n + 4) * work_dim1 + 1], &work[(*n + 6) * work_dim1 + 1], &ierr); } } goto L50; } } sep[ks] = scale / max(est,smlnum); if (pair) { sep[ks + 1] = sep[ks]; } } if (pair) { ++ks; } L60: ; } return 0; /* End of DTRSNA */ }
/* Subroutine */ int dtrsna_(char *job, char *howmny, logical *select, integer *n, doublereal *t, integer *ldt, doublereal *vl, integer * ldvl, doublereal *vr, integer *ldvr, doublereal *s, doublereal *sep, integer *mm, integer *m, doublereal *work, integer *ldwork, integer * iwork, integer *info) { /* -- LAPACK routine (version 2.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 Purpose ======= DTRSNA estimates reciprocal condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix T (or of any matrix Q*T*Q**T with Q orthogonal). T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. Arguments ========= JOB (input) CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (SEP): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (SEP); = 'B': for both eigenvalues and eigenvectors (S and SEP). HOWMNY (input) CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. SELECT (input) LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the eigenpair corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues w(j) and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be set to .TRUE.. If HOWMNY = 'A', SELECT is not referenced. N (input) INTEGER The order of the matrix T. N >= 0. T (input) DOUBLE PRECISION array, dimension (LDT,N) The upper quasi-triangular matrix T, in Schur canonical form. LDT (input) INTEGER The leading dimension of the array T. LDT >= max(1,N). VL (input) DOUBLE PRECISION array, dimension (LDVL,M) If JOB = 'E' or 'B', VL must contain left eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VL, as returned by DHSEIN or DTREVC. If JOB = 'V', VL is not referenced. LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= 1; and if JOB = 'E' or 'B', LDVL >= N. VR (input) DOUBLE PRECISION array, dimension (LDVR,M) If JOB = 'E' or 'B', VR must contain right eigenvectors of T (or of any Q*T*Q**T with Q orthogonal), corresponding to the eigenpairs specified by HOWMNY and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by DHSEIN or DTREVC. If JOB = 'V', VR is not referenced. LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= 1; and if JOB = 'E' or 'B', LDVR >= N. S (output) DOUBLE PRECISION array, dimension (MM) If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of S are set to the same value. Thus S(j), SEP(j), and the j-th columns of VL and VR all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected). If JOB = 'V', S is not referenced. SEP (output) DOUBLE PRECISION array, dimension (MM) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of SEP are set to the same value. If the eigenvalues cannot be reordered to compute SEP(j), SEP(j) is set to 0; this can only occur when the true value would be very small anyway. If JOB = 'E', SEP is not referenced. MM (input) INTEGER The number of elements in the arrays S (if JOB = 'E' or 'B') and/or SEP (if JOB = 'V' or 'B'). MM >= M. M (output) INTEGER The number of elements of the arrays S and/or SEP actually used to store the estimated condition numbers. If HOWMNY = 'A', M is set to N. WORK (workspace) DOUBLE PRECISION array, dimension (LDWORK,N+1) If JOB = 'E', WORK is not referenced. LDWORK (input) INTEGER The leading dimension of the array WORK. LDWORK >= 1; and if JOB = 'V' or 'B', LDWORK >= N. IWORK (workspace) INTEGER array, dimension (N) If JOB = 'E', IWORK is not referenced. INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value Further Details =============== The reciprocal of the condition number of an eigenvalue lambda is defined as S(lambda) = |v'*u| / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of T corresponding to lambda; v' denotes the conjugate-transpose of v, and norm(u) denotes the Euclidean norm. These reciprocal condition numbers always lie between zero (very badly conditioned) and one (very well conditioned). If n = 1, S(lambda) is defined to be 1. An approximate error bound for a computed eigenvalue W(i) is given by EPS * norm(T) / S(i) where EPS is the machine precision. The reciprocal of the condition number of the right eigenvector u corresponding to lambda is defined as follows. Suppose T = ( lambda c ) ( 0 T22 ) Then the reciprocal condition number is SEP( lambda, T22 ) = sigma-min( T22 - lambda*I ) where sigma-min denotes the smallest singular value. We approximate the smallest singular value by the reciprocal of an estimate of the one-norm of the inverse of T22 - lambda*I. If n = 1, SEP(1) is defined to be abs(T(1,1)). An approximate error bound for a computed right eigenvector VR(i) is given by EPS * norm(T) / SEP(i) ===================================================================== Decode and test the input parameters Parameter adjustments Function Body */ /* Table of constant values */ static integer c__1 = 1; static logical c_true = TRUE_; static logical c_false = FALSE_; /* System generated locals */ integer t_dim1, t_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, work_dim1, work_offset, i__1, i__2; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ static integer kase; static doublereal cond; extern doublereal ddot_(integer *, doublereal *, integer *, doublereal *, integer *); static logical pair; static integer ierr; static doublereal dumm, prod; static integer ifst; static doublereal lnrm; static integer ilst; static doublereal rnrm; extern doublereal dnrm2_(integer *, doublereal *, integer *); static doublereal prod1, prod2; static integer i, j, k; static doublereal scale, delta; extern logical lsame_(char *, char *); static logical wants; static doublereal dummy[1]; static integer n2; extern doublereal dlapy2_(doublereal *, doublereal *); extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); static doublereal cs; extern doublereal dlamch_(char *); static integer nn, ks; extern /* Subroutine */ int dlacon_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static doublereal sn, mu; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *); static doublereal bignum; static logical wantbh; extern /* Subroutine */ int dlaqtr_(logical *, logical *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *), dtrexc_(char *, integer * , doublereal *, integer *, doublereal *, integer *, integer *, integer *, doublereal *, integer *); static logical somcon; static doublereal smlnum; static logical wantsp; static doublereal eps, est; #define DUMMY(I) dummy[(I)] #define SELECT(I) select[(I)-1] #define S(I) s[(I)-1] #define SEP(I) sep[(I)-1] #define IWORK(I) iwork[(I)-1] #define T(I,J) t[(I)-1 + ((J)-1)* ( *ldt)] #define VL(I,J) vl[(I)-1 + ((J)-1)* ( *ldvl)] #define VR(I,J) vr[(I)-1 + ((J)-1)* ( *ldvr)] #define WORK(I,J) work[(I)-1 + ((J)-1)* ( *ldwork)] wantbh = lsame_(job, "B"); wants = lsame_(job, "E") || wantbh; wantsp = lsame_(job, "V") || wantbh; somcon = lsame_(howmny, "S"); *info = 0; if (! wants && ! wantsp) { *info = -1; } else if (! lsame_(howmny, "A") && ! somcon) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*ldt < max(1,*n)) { *info = -6; } else if (*ldvl < 1 || wants && *ldvl < *n) { *info = -8; } else if (*ldvr < 1 || wants && *ldvr < *n) { *info = -10; } else { /* Set M to the number of eigenpairs for which condition number s are required, and test MM. */ if (somcon) { *m = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= *n; ++k) { if (pair) { pair = FALSE_; } else { if (k < *n) { if (T(k+1,k) == 0.) { if (SELECT(k)) { ++(*m); } } else { pair = TRUE_; if (SELECT(k) || SELECT(k + 1)) { *m += 2; } } } else { if (SELECT(*n)) { ++(*m); } } } /* L10: */ } } else { *m = *n; } if (*mm < *m) { *info = -13; } else if (*ldwork < 1 || wantsp && *ldwork < *n) { *info = -16; } } if (*info != 0) { i__1 = -(*info); xerbla_("DTRSNA", &i__1); return 0; } /* Quick return if possible */ if (*n == 0) { return 0; } if (*n == 1) { if (somcon) { if (! SELECT(1)) { return 0; } } if (wants) { S(1) = 1.; } if (wantsp) { SEP(1) = (d__1 = T(1,1), abs(d__1)); } return 0; } /* Get machine constants */ eps = dlamch_("P"); smlnum = dlamch_("S") / eps; bignum = 1. / smlnum; dlabad_(&smlnum, &bignum); ks = 0; pair = FALSE_; i__1 = *n; for (k = 1; k <= *n; ++k) { /* Determine whether T(k,k) begins a 1-by-1 or 2-by-2 block. */ if (pair) { pair = FALSE_; goto L60; } else { if (k < *n) { pair = T(k+1,k) != 0.; } } /* Determine whether condition numbers are required for the k-t h eigenpair. */ if (somcon) { if (pair) { if (! SELECT(k) && ! SELECT(k + 1)) { goto L60; } } else { if (! SELECT(k)) { goto L60; } } } ++ks; if (wants) { /* Compute the reciprocal condition number of the k-th eigenvalue. */ if (! pair) { /* Real eigenvalue. */ prod = ddot_(n, &VR(1,ks), &c__1, &VL(1,ks), &c__1); rnrm = dnrm2_(n, &VR(1,ks), &c__1); lnrm = dnrm2_(n, &VL(1,ks), &c__1); S(ks) = abs(prod) / (rnrm * lnrm); } else { /* Complex eigenvalue. */ prod1 = ddot_(n, &VR(1,ks), &c__1, &VL(1,ks), &c__1); prod1 += ddot_(n, &VR(1,ks+1), &c__1, &VL(1,ks+1), &c__1); prod2 = ddot_(n, &VL(1,ks), &c__1, &VR(1,ks+1), &c__1); prod2 -= ddot_(n, &VL(1,ks+1), &c__1, &VR(1,ks), &c__1); d__1 = dnrm2_(n, &VR(1,ks), &c__1); d__2 = dnrm2_(n, &VR(1,ks+1), &c__1); rnrm = dlapy2_(&d__1, &d__2); d__1 = dnrm2_(n, &VL(1,ks), &c__1); d__2 = dnrm2_(n, &VL(1,ks+1), &c__1); lnrm = dlapy2_(&d__1, &d__2); cond = dlapy2_(&prod1, &prod2) / (rnrm * lnrm); S(ks) = cond; S(ks + 1) = cond; } } if (wantsp) { /* Estimate the reciprocal condition number of the k-th eigenvector. Copy the matrix T to the array WORK and swap the diag onal block beginning at T(k,k) to the (1,1) position. */ dlacpy_("Full", n, n, &T(1,1), ldt, &WORK(1,1), ldwork); ifst = k; ilst = 1; dtrexc_("No Q", n, &WORK(1,1), ldwork, dummy, &c__1, & ifst, &ilst, &WORK(1,*n+1), &ierr); if (ierr == 1 || ierr == 2) { /* Could not swap because blocks not well separat ed */ scale = 1.; est = bignum; } else { /* Reordering successful */ if (WORK(2,1) == 0.) { /* Form C = T22 - lambda*I in WORK(2:N,2:N ). */ i__2 = *n; for (i = 2; i <= *n; ++i) { WORK(i,i) -= WORK(1,1); /* L20: */ } n2 = 1; nn = *n - 1; } else { /* Triangularize the 2 by 2 block by unita ry transformation U = [ cs i*ss ] [ i*ss cs ]. such that the (1,1) position of WORK is complex eigenvalue lambda with positive imagina ry part. (2,2) position of WORK is the complex eigenva lue lambda with negative imaginary part. */ mu = sqrt((d__1 = WORK(1,2), abs(d__1))) * sqrt((d__2 = WORK(2,1), abs(d__2))); delta = dlapy2_(&mu, &WORK(2,1)); cs = mu / delta; sn = -WORK(2,1) / delta; /* Form C' = WORK(2:N,2:N) + i*[rwork(1) ..... rwork(n-1) ] [ mu ] [ .. ] [ .. ] [ mu ] where C' is conjugate transpose of comp lex matrix C, and RWORK is stored starting in the N+1 -st column of WORK. */ i__2 = *n; for (j = 3; j <= *n; ++j) { WORK(2,j) = cs * WORK(2,j) ; WORK(j,j) -= WORK(1,1); /* L30: */ } WORK(2,2) = 0.; WORK(1,*n+1) = mu * 2.; i__2 = *n - 1; for (i = 2; i <= *n-1; ++i) { WORK(i,*n+1) = sn * WORK(1,i+1); /* L40: */ } n2 = 2; nn = *n - 1 << 1; } /* Estimate norm(inv(C')) */ est = 0.; kase = 0; L50: dlacon_(&nn, &WORK(1,*n+2), &WORK(1,*n+4), &IWORK(1), &est, &kase); if (kase != 0) { if (kase == 1) { if (n2 == 1) { /* Real eigenvalue: solve C' *x = scale*c. */ i__2 = *n - 1; dlaqtr_(&c_true, &c_true, &i__2, &WORK(2,2), ldwork, dummy, &dumm, &scale, &WORK(1,*n+4), &WORK(1,*n+6), &ierr); } else { /* Complex eigenvalue: solve C'*(p+iq) = scale*(c+id) in real arithmetic. */ i__2 = *n - 1; dlaqtr_(&c_true, &c_false, &i__2, &WORK(2,2), ldwork, &WORK(1,*n+1), &mu, &scale, &WORK(1,*n+4), &WORK(1,*n+6), &ierr); } } else { if (n2 == 1) { /* Real eigenvalue: solve C* x = scale*c. */ i__2 = *n - 1; dlaqtr_(&c_false, &c_true, &i__2, &WORK(2,2), ldwork, dummy, & dumm, &scale, &WORK(1,*n+4), &WORK(1,*n+6), & ierr); } else { /* Complex eigenvalue: solve C*(p+iq) = scale*(c+id) i n real arithmetic. */ i__2 = *n - 1; dlaqtr_(&c_false, &c_false, &i__2, &WORK(2,2), ldwork, &WORK(1,*n+1), &mu, &scale, &WORK(1,*n+4), &WORK(1,*n+6), &ierr); } } goto L50; } } SEP(ks) = scale / max(est,smlnum); if (pair) { SEP(ks + 1) = SEP(ks); } } if (pair) { ++ks; } L60: ; } return 0; /* End of DTRSNA */ } /* dtrsna_ */