#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dtgevc_(char *side, char *howmny, logical *select, integer *n, doublereal *a, integer *lda, doublereal *b, integer *ldb, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, integer *mm, integer *m, doublereal *work, integer *info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University June 30, 1999 Purpose ======= DTGEVC computes some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B). The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by: (A - wB) * x = 0 and y**H * (A - wB) = 0 where y**H denotes the conjugate tranpose of y. If an eigenvalue w is determined by zero diagonal elements of both A and B, a unit vector is returned as the corresponding eigenvector. If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of (A,B), or the products Z*X and/or Q*Y, where Z and Q are input orthogonal matrices. If (A,B) was obtained from the generalized real-Schur factorization of an original pair of matrices (A0,B0) = (Q*A*Z**H,Q*B*Z**H), then Z*X and Q*Y are the matrices of right or left eigenvectors of A. A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks. Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part. Arguments ========= SIDE (input) CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. HOWMNY (input) CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'B': compute all right and/or left eigenvectors, and backtransform them using the input matrices supplied in VR and/or VL; = 'S': compute selected right and/or left eigenvectors, specified by the logical array SELECT. SELECT (input) LOGICAL array, dimension (N) If HOWMNY='S', SELECT specifies the eigenvectors to be computed. If HOWMNY='A' or 'B', SELECT is not referenced. To select the real eigenvector corresponding to the real eigenvalue w(j), SELECT(j) must be set to .TRUE. To select the complex eigenvector corresponding to a complex conjugate pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be set to .TRUE.. N (input) INTEGER The order of the matrices A and B. N >= 0. A (input) DOUBLE PRECISION array, dimension (LDA,N) The upper quasi-triangular matrix A. LDA (input) INTEGER The leading dimension of array A. LDA >= max(1, N). B (input) DOUBLE PRECISION array, dimension (LDB,N) The upper triangular matrix B. If A has a 2-by-2 diagonal block, then the corresponding 2-by-2 block of B must be diagonal with positive elements. LDB (input) INTEGER The leading dimension of array B. LDB >= max(1,N). VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM) On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must contain an N-by-N matrix Q (usually the orthogonal matrix Q of left Schur vectors returned by DHGEQZ). On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of (A,B); if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of (A,B) specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues. If SIDE = 'R', VL is not referenced. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. LDVL (input) INTEGER The leading dimension of array VL. LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise. VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM) On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an N-by-N matrix Q (usually the orthogonal matrix Z of right Schur vectors returned by DHGEQZ). On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of (A,B); if HOWMNY = 'B', the matrix Z*X; if HOWMNY = 'S', the right eigenvectors of (A,B) specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues. If SIDE = 'L', VR is not referenced. A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part. LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise. MM (input) INTEGER The number of columns in the arrays VL and/or VR. MM >= M. M (output) INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = 'A' or 'B', M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns. WORK (workspace) DOUBLE PRECISION array, dimension (6*N) INFO (output) INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value. > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex eigenvalue. Further Details =============== Allocation of workspace: ---------- -- --------- WORK( j ) = 1-norm of j-th column of A, above the diagonal WORK( N+j ) = 1-norm of j-th column of B, above the diagonal WORK( 2*N+1:3*N ) = real part of eigenvector WORK( 3*N+1:4*N ) = imaginary part of eigenvector WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector Rowwise vs. columnwise solution methods: ------- -- ---------- -------- ------- Finding a generalized eigenvector consists basically of solving the singular triangular system (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left) Consider finding the i-th right eigenvector (assume all eigenvalues are real). The equation to be solved is: n i 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1 k=j k=j where C = (A - w B) (The components v(i+1:n) are 0.) The "rowwise" method is: (1) v(i) := 1 for j = i-1,. . .,1: i (2) compute s = - sum C(j,k) v(k) and k=j+1 (3) v(j) := s / C(j,j) Step 2 is sometimes called the "dot product" step, since it is an inner product between the j-th row and the portion of the eigenvector that has been computed so far. The "columnwise" method consists basically in doing the sums for all the rows in parallel. As each v(j) is computed, the contribution of v(j) times the j-th column of C is added to the partial sums. Since FORTRAN arrays are stored columnwise, this has the advantage that at each step, the elements of C that are accessed are adjacent to one another, whereas with the rowwise method, the elements accessed at a step are spaced LDA (and LDB) words apart. When finding left eigenvectors, the matrix in question is the transpose of the one in storage, so the rowwise method then actually accesses columns of A and B at each step, and so is the preferred method. ===================================================================== Decode and Test the input parameters Parameter adjustments */ /* Table of constant values */ static logical c_true = TRUE_; static integer c__2 = 2; static doublereal c_b35 = 1.; static integer c__1 = 1; static doublereal c_b37 = 0.; static logical c_false = FALSE_; /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset, vl_dim1, vl_offset, vr_dim1, vr_offset, i__1, i__2, i__3, i__4, i__5; doublereal d__1, d__2, d__3, d__4, d__5, d__6; /* Local variables */ static integer ibeg, ieig, iend; static doublereal dmin__, temp, suma[4] /* was [2][2] */, sumb[4] /* was [2][2] */, xmax; extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); static doublereal cim2a, cim2b, cre2a, cre2b, temp2, bdiag[2]; static integer i__, j; static doublereal acoef, scale; static logical ilall; static integer iside; static doublereal sbeta; extern logical lsame_(char *, char *); extern /* Subroutine */ int dgemv_(char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); static logical il2by2; static integer iinfo; static doublereal small; static logical compl; static doublereal anorm, bnorm; static logical compr; extern /* Subroutine */ int dlaln2_(logical *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal * , doublereal *, integer *, doublereal *, doublereal *, integer *); static doublereal temp2i; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); static doublereal temp2r; static integer ja; static logical ilabad, ilbbad; static integer jc, je, na; static doublereal acoefa, bcoefa, cimaga, cimagb; static logical ilback; static integer im; static doublereal bcoefi, ascale, bscale, creala; static integer jr; static doublereal crealb; extern doublereal dlamch_(char *); static doublereal bcoefr; static integer jw, nw; static doublereal salfar, safmin; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); static doublereal xscale, bignum; extern /* Subroutine */ int xerbla_(char *, integer *); static logical ilcomp, ilcplx; static integer ihwmny; static doublereal big; static logical lsa, lsb; static doublereal ulp, sum[4] /* was [2][2] */; #define suma_ref(a_1,a_2) suma[(a_2)*2 + a_1 - 3] #define sumb_ref(a_1,a_2) sumb[(a_2)*2 + a_1 - 3] #define a_ref(a_1,a_2) a[(a_2)*a_dim1 + a_1] #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] #define vl_ref(a_1,a_2) vl[(a_2)*vl_dim1 + a_1] #define vr_ref(a_1,a_2) vr[(a_2)*vr_dim1 + a_1] #define sum_ref(a_1,a_2) sum[(a_2)*2 + a_1 - 3] --select; a_dim1 = *lda; a_offset = 1 + a_dim1 * 1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; vl_dim1 = *ldvl; vl_offset = 1 + vl_dim1 * 1; vl -= vl_offset; vr_dim1 = *ldvr; vr_offset = 1 + vr_dim1 * 1; vr -= vr_offset; --work; /* Function Body */ if (lsame_(howmny, "A")) { ihwmny = 1; ilall = TRUE_; ilback = FALSE_; } else if (lsame_(howmny, "S")) { ihwmny = 2; ilall = FALSE_; ilback = FALSE_; } else if (lsame_(howmny, "B") || lsame_(howmny, "T")) { ihwmny = 3; ilall = TRUE_; ilback = TRUE_; } else { ihwmny = -1; ilall = TRUE_; } if (lsame_(side, "R")) { iside = 1; compl = FALSE_; compr = TRUE_; } else if (lsame_(side, "L")) { iside = 2; compl = TRUE_; compr = FALSE_; } else if (lsame_(side, "B")) { iside = 3; compl = TRUE_; compr = TRUE_; } else { iside = -1; } *info = 0; if (iside < 0) { *info = -1; } else if (ihwmny < 0) { *info = -2; } else if (*n < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldb < max(1,*n)) { *info = -8; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGEVC", &i__1); return 0; } /* Count the number of eigenvectors to be computed */ if (! ilall) { im = 0; ilcplx = FALSE_; i__1 = *n; for (j = 1; j <= i__1; ++j) { if (ilcplx) { ilcplx = FALSE_; goto L10; } if (j < *n) { if (a_ref(j + 1, j) != 0.) { ilcplx = TRUE_; } } if (ilcplx) { if (select[j] || select[j + 1]) { im += 2; } } else { if (select[j]) { ++im; } } L10: ; } } else { im = *n; } /* Check 2-by-2 diagonal blocks of A, B */ ilabad = FALSE_; ilbbad = FALSE_; i__1 = *n - 1; for (j = 1; j <= i__1; ++j) { if (a_ref(j + 1, j) != 0.) { if (b_ref(j, j) == 0. || b_ref(j + 1, j + 1) == 0. || b_ref(j, j + 1) != 0.) { ilbbad = TRUE_; } if (j < *n - 1) { if (a_ref(j + 2, j + 1) != 0.) { ilabad = TRUE_; } } } /* L20: */ } if (ilabad) { *info = -5; } else if (ilbbad) { *info = -7; } else if (compl && *ldvl < *n || *ldvl < 1) { *info = -10; } else if (compr && *ldvr < *n || *ldvr < 1) { *info = -12; } else if (*mm < im) { *info = -13; } if (*info != 0) { i__1 = -(*info); xerbla_("DTGEVC", &i__1); return 0; } /* Quick return if possible */ *m = im; if (*n == 0) { return 0; } /* Machine Constants */ safmin = dlamch_("Safe minimum"); big = 1. / safmin; dlabad_(&safmin, &big); ulp = dlamch_("Epsilon") * dlamch_("Base"); small = safmin * *n / ulp; big = 1. / small; bignum = 1. / (safmin * *n); /* Compute the 1-norm of each column of the strictly upper triangular part (i.e., excluding all elements belonging to the diagonal blocks) of A and B to check for possible overflow in the triangular solver. */ anorm = (d__1 = a_ref(1, 1), abs(d__1)); if (*n > 1) { anorm += (d__1 = a_ref(2, 1), abs(d__1)); } bnorm = (d__1 = b_ref(1, 1), abs(d__1)); work[1] = 0.; work[*n + 1] = 0.; i__1 = *n; for (j = 2; j <= i__1; ++j) { temp = 0.; temp2 = 0.; if (a_ref(j, j - 1) == 0.) { iend = j - 1; } else { iend = j - 2; } i__2 = iend; for (i__ = 1; i__ <= i__2; ++i__) { temp += (d__1 = a_ref(i__, j), abs(d__1)); temp2 += (d__1 = b_ref(i__, j), abs(d__1)); /* L30: */ } work[j] = temp; work[*n + j] = temp2; /* Computing MIN */ i__3 = j + 1; i__2 = min(i__3,*n); for (i__ = iend + 1; i__ <= i__2; ++i__) { temp += (d__1 = a_ref(i__, j), abs(d__1)); temp2 += (d__1 = b_ref(i__, j), abs(d__1)); /* L40: */ } anorm = max(anorm,temp); bnorm = max(bnorm,temp2); /* L50: */ } ascale = 1. / max(anorm,safmin); bscale = 1. / max(bnorm,safmin); /* Left eigenvectors */ if (compl) { ieig = 0; /* Main loop over eigenvalues */ ilcplx = FALSE_; i__1 = *n; for (je = 1; je <= i__1; ++je) { /* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or (b) this would be the second of a complex pair. Check for complex eigenvalue, so as to be sure of which entry(-ies) of SELECT to look at. */ if (ilcplx) { ilcplx = FALSE_; goto L220; } nw = 1; if (je < *n) { if (a_ref(je + 1, je) != 0.) { ilcplx = TRUE_; nw = 2; } } if (ilall) { ilcomp = TRUE_; } else if (ilcplx) { ilcomp = select[je] || select[je + 1]; } else { ilcomp = select[je]; } if (! ilcomp) { goto L220; } /* Decide if (a) singular pencil, (b) real eigenvalue, or (c) complex eigenvalue. */ if (! ilcplx) { if ((d__1 = a_ref(je, je), abs(d__1)) <= safmin && (d__2 = b_ref(je, je), abs(d__2)) <= safmin) { /* Singular matrix pencil -- return unit eigenvector */ ++ieig; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl_ref(jr, ieig) = 0.; /* L60: */ } vl_ref(ieig, ieig) = 1.; goto L220; } } /* Clear vector */ i__2 = nw * *n; for (jr = 1; jr <= i__2; ++jr) { work[(*n << 1) + jr] = 0.; /* L70: */ } /* T Compute coefficients in ( a A - b B ) y = 0 a is ACOEF b is BCOEFR + i*BCOEFI */ if (! ilcplx) { /* Real eigenvalue Computing MAX */ d__3 = (d__1 = a_ref(je, je), abs(d__1)) * ascale, d__4 = ( d__2 = b_ref(je, je), abs(d__2)) * bscale, d__3 = max( d__3,d__4); temp = 1. / max(d__3,safmin); salfar = temp * a_ref(je, je) * ascale; sbeta = temp * b_ref(je, je) * bscale; acoef = sbeta * ascale; bcoefr = salfar * bscale; bcoefi = 0.; /* Scale to avoid underflow */ scale = 1.; lsa = abs(sbeta) >= safmin && abs(acoef) < small; lsb = abs(salfar) >= safmin && abs(bcoefr) < small; if (lsa) { scale = small / abs(sbeta) * min(anorm,big); } if (lsb) { /* Computing MAX */ d__1 = scale, d__2 = small / abs(salfar) * min(bnorm,big); scale = max(d__1,d__2); } if (lsa || lsb) { /* Computing MIN Computing MAX */ d__3 = 1., d__4 = abs(acoef), d__3 = max(d__3,d__4), d__4 = abs(bcoefr); d__1 = scale, d__2 = 1. / (safmin * max(d__3,d__4)); scale = min(d__1,d__2); if (lsa) { acoef = ascale * (scale * sbeta); } else { acoef = scale * acoef; } if (lsb) { bcoefr = bscale * (scale * salfar); } else { bcoefr = scale * bcoefr; } } acoefa = abs(acoef); bcoefa = abs(bcoefr); /* First component is 1 */ work[(*n << 1) + je] = 1.; xmax = 1.; } else { /* Complex eigenvalue */ d__1 = safmin * 100.; dlag2_(&a_ref(je, je), lda, &b_ref(je, je), ldb, &d__1, & acoef, &temp, &bcoefr, &temp2, &bcoefi); bcoefi = -bcoefi; if (bcoefi == 0.) { *info = je; return 0; } /* Scale to avoid over/underflow */ acoefa = abs(acoef); bcoefa = abs(bcoefr) + abs(bcoefi); scale = 1.; if (acoefa * ulp < safmin && acoefa >= safmin) { scale = safmin / ulp / acoefa; } if (bcoefa * ulp < safmin && bcoefa >= safmin) { /* Computing MAX */ d__1 = scale, d__2 = safmin / ulp / bcoefa; scale = max(d__1,d__2); } if (safmin * acoefa > ascale) { scale = ascale / (safmin * acoefa); } if (safmin * bcoefa > bscale) { /* Computing MIN */ d__1 = scale, d__2 = bscale / (safmin * bcoefa); scale = min(d__1,d__2); } if (scale != 1.) { acoef = scale * acoef; acoefa = abs(acoef); bcoefr = scale * bcoefr; bcoefi = scale * bcoefi; bcoefa = abs(bcoefr) + abs(bcoefi); } /* Compute first two components of eigenvector */ temp = acoef * a_ref(je + 1, je); temp2r = acoef * a_ref(je, je) - bcoefr * b_ref(je, je); temp2i = -bcoefi * b_ref(je, je); if (abs(temp) > abs(temp2r) + abs(temp2i)) { work[(*n << 1) + je] = 1.; work[*n * 3 + je] = 0.; work[(*n << 1) + je + 1] = -temp2r / temp; work[*n * 3 + je + 1] = -temp2i / temp; } else { work[(*n << 1) + je + 1] = 1.; work[*n * 3 + je + 1] = 0.; temp = acoef * a_ref(je, je + 1); work[(*n << 1) + je] = (bcoefr * b_ref(je + 1, je + 1) - acoef * a_ref(je + 1, je + 1)) / temp; work[*n * 3 + je] = bcoefi * b_ref(je + 1, je + 1) / temp; } /* Computing MAX */ d__5 = (d__1 = work[(*n << 1) + je], abs(d__1)) + (d__2 = work[*n * 3 + je], abs(d__2)), d__6 = (d__3 = work[(* n << 1) + je + 1], abs(d__3)) + (d__4 = work[*n * 3 + je + 1], abs(d__4)); xmax = max(d__5,d__6); } /* Computing MAX */ d__1 = ulp * acoefa * anorm, d__2 = ulp * bcoefa * bnorm, d__1 = max(d__1,d__2); dmin__ = max(d__1,safmin); /* T Triangular solve of (a A - b B) y = 0 T (rowwise in (a A - b B) , or columnwise in (a A - b B) ) */ il2by2 = FALSE_; i__2 = *n; for (j = je + nw; j <= i__2; ++j) { if (il2by2) { il2by2 = FALSE_; goto L160; } na = 1; bdiag[0] = b_ref(j, j); if (j < *n) { if (a_ref(j + 1, j) != 0.) { il2by2 = TRUE_; bdiag[1] = b_ref(j + 1, j + 1); na = 2; } } /* Check whether scaling is necessary for dot products */ xscale = 1. / max(1.,xmax); /* Computing MAX */ d__1 = work[j], d__2 = work[*n + j], d__1 = max(d__1,d__2), d__2 = acoefa * work[j] + bcoefa * work[*n + j]; temp = max(d__1,d__2); if (il2by2) { /* Computing MAX */ d__1 = temp, d__2 = work[j + 1], d__1 = max(d__1,d__2), d__2 = work[*n + j + 1], d__1 = max(d__1,d__2), d__2 = acoefa * work[j + 1] + bcoefa * work[*n + j + 1]; temp = max(d__1,d__2); } if (temp > bignum * xscale) { i__3 = nw - 1; for (jw = 0; jw <= i__3; ++jw) { i__4 = j - 1; for (jr = je; jr <= i__4; ++jr) { work[(jw + 2) * *n + jr] = xscale * work[(jw + 2) * *n + jr]; /* L80: */ } /* L90: */ } xmax *= xscale; } /* Compute dot products j-1 SUM = sum conjg( a*A(k,j) - b*B(k,j) )*x(k) k=je To reduce the op count, this is done as _ j-1 _ j-1 a*conjg( sum A(k,j)*x(k) ) - b*conjg( sum B(k,j)*x(k) ) k=je k=je which may cause underflow problems if A or B are close to underflow. (E.g., less than SMALL.) A series of compiler directives to defeat vectorization for the next loop $PL$ CMCHAR=' ' DIR$ NEXTSCALAR $DIR SCALAR DIR$ NEXT SCALAR VD$L NOVECTOR DEC$ NOVECTOR VD$ NOVECTOR VDIR NOVECTOR VOCL LOOP,SCALAR IBM PREFER SCALAR $PL$ CMCHAR='*' */ i__3 = nw; for (jw = 1; jw <= i__3; ++jw) { /* $PL$ CMCHAR=' ' DIR$ NEXTSCALAR $DIR SCALAR DIR$ NEXT SCALAR VD$L NOVECTOR DEC$ NOVECTOR VD$ NOVECTOR VDIR NOVECTOR VOCL LOOP,SCALAR IBM PREFER SCALAR $PL$ CMCHAR='*' */ i__4 = na; for (ja = 1; ja <= i__4; ++ja) { suma_ref(ja, jw) = 0.; sumb_ref(ja, jw) = 0.; i__5 = j - 1; for (jr = je; jr <= i__5; ++jr) { suma_ref(ja, jw) = suma_ref(ja, jw) + a_ref(jr, j + ja - 1) * work[(jw + 1) * *n + jr]; sumb_ref(ja, jw) = sumb_ref(ja, jw) + b_ref(jr, j + ja - 1) * work[(jw + 1) * *n + jr]; /* L100: */ } /* L110: */ } /* L120: */ } /* $PL$ CMCHAR=' ' DIR$ NEXTSCALAR $DIR SCALAR DIR$ NEXT SCALAR VD$L NOVECTOR DEC$ NOVECTOR VD$ NOVECTOR VDIR NOVECTOR VOCL LOOP,SCALAR IBM PREFER SCALAR $PL$ CMCHAR='*' */ i__3 = na; for (ja = 1; ja <= i__3; ++ja) { if (ilcplx) { sum_ref(ja, 1) = -acoef * suma_ref(ja, 1) + bcoefr * sumb_ref(ja, 1) - bcoefi * sumb_ref(ja, 2); sum_ref(ja, 2) = -acoef * suma_ref(ja, 2) + bcoefr * sumb_ref(ja, 2) + bcoefi * sumb_ref(ja, 1); } else { sum_ref(ja, 1) = -acoef * suma_ref(ja, 1) + bcoefr * sumb_ref(ja, 1); } /* L130: */ } /* T Solve ( a A - b B ) y = SUM(,) with scaling and perturbation of the denominator */ dlaln2_(&c_true, &na, &nw, &dmin__, &acoef, &a_ref(j, j), lda, bdiag, &bdiag[1], sum, &c__2, &bcoefr, &bcoefi, & work[(*n << 1) + j], n, &scale, &temp, &iinfo); if (scale < 1.) { i__3 = nw - 1; for (jw = 0; jw <= i__3; ++jw) { i__4 = j - 1; for (jr = je; jr <= i__4; ++jr) { work[(jw + 2) * *n + jr] = scale * work[(jw + 2) * *n + jr]; /* L140: */ } /* L150: */ } xmax = scale * xmax; } xmax = max(xmax,temp); L160: ; } /* Copy eigenvector to VL, back transforming if HOWMNY='B'. */ ++ieig; if (ilback) { i__2 = nw - 1; for (jw = 0; jw <= i__2; ++jw) { i__3 = *n + 1 - je; dgemv_("N", n, &i__3, &c_b35, &vl_ref(1, je), ldvl, &work[ (jw + 2) * *n + je], &c__1, &c_b37, &work[(jw + 4) * *n + 1], &c__1); /* L170: */ } dlacpy_(" ", n, &nw, &work[(*n << 2) + 1], n, &vl_ref(1, je), ldvl); ibeg = 1; } else { dlacpy_(" ", n, &nw, &work[(*n << 1) + 1], n, &vl_ref(1, ieig) , ldvl); ibeg = je; } /* Scale eigenvector */ xmax = 0.; if (ilcplx) { i__2 = *n; for (j = ibeg; j <= i__2; ++j) { /* Computing MAX */ d__3 = xmax, d__4 = (d__1 = vl_ref(j, ieig), abs(d__1)) + (d__2 = vl_ref(j, ieig + 1), abs(d__2)); xmax = max(d__3,d__4); /* L180: */ } } else { i__2 = *n; for (j = ibeg; j <= i__2; ++j) { /* Computing MAX */ d__2 = xmax, d__3 = (d__1 = vl_ref(j, ieig), abs(d__1)); xmax = max(d__2,d__3); /* L190: */ } } if (xmax > safmin) { xscale = 1. / xmax; i__2 = nw - 1; for (jw = 0; jw <= i__2; ++jw) { i__3 = *n; for (jr = ibeg; jr <= i__3; ++jr) { vl_ref(jr, ieig + jw) = xscale * vl_ref(jr, ieig + jw) ; /* L200: */ } /* L210: */ } } ieig = ieig + nw - 1; L220: ; } } /* Right eigenvectors */ if (compr) { ieig = im + 1; /* Main loop over eigenvalues */ ilcplx = FALSE_; for (je = *n; je >= 1; --je) { /* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or (b) this would be the second of a complex pair. Check for complex eigenvalue, so as to be sure of which entry(-ies) of SELECT to look at -- if complex, SELECT(JE) or SELECT(JE-1). If this is a complex pair, the 2-by-2 diagonal block corresponding to the eigenvalue is in rows/columns JE-1:JE */ if (ilcplx) { ilcplx = FALSE_; goto L500; } nw = 1; if (je > 1) { if (a_ref(je, je - 1) != 0.) { ilcplx = TRUE_; nw = 2; } } if (ilall) { ilcomp = TRUE_; } else if (ilcplx) { ilcomp = select[je] || select[je - 1]; } else { ilcomp = select[je]; } if (! ilcomp) { goto L500; } /* Decide if (a) singular pencil, (b) real eigenvalue, or (c) complex eigenvalue. */ if (! ilcplx) { if ((d__1 = a_ref(je, je), abs(d__1)) <= safmin && (d__2 = b_ref(je, je), abs(d__2)) <= safmin) { /* Singular matrix pencil -- unit eigenvector */ --ieig; i__1 = *n; for (jr = 1; jr <= i__1; ++jr) { vr_ref(jr, ieig) = 0.; /* L230: */ } vr_ref(ieig, ieig) = 1.; goto L500; } } /* Clear vector */ i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { work[(jw + 2) * *n + jr] = 0.; /* L240: */ } /* L250: */ } /* Compute coefficients in ( a A - b B ) x = 0 a is ACOEF b is BCOEFR + i*BCOEFI */ if (! ilcplx) { /* Real eigenvalue Computing MAX */ d__3 = (d__1 = a_ref(je, je), abs(d__1)) * ascale, d__4 = ( d__2 = b_ref(je, je), abs(d__2)) * bscale, d__3 = max( d__3,d__4); temp = 1. / max(d__3,safmin); salfar = temp * a_ref(je, je) * ascale; sbeta = temp * b_ref(je, je) * bscale; acoef = sbeta * ascale; bcoefr = salfar * bscale; bcoefi = 0.; /* Scale to avoid underflow */ scale = 1.; lsa = abs(sbeta) >= safmin && abs(acoef) < small; lsb = abs(salfar) >= safmin && abs(bcoefr) < small; if (lsa) { scale = small / abs(sbeta) * min(anorm,big); } if (lsb) { /* Computing MAX */ d__1 = scale, d__2 = small / abs(salfar) * min(bnorm,big); scale = max(d__1,d__2); } if (lsa || lsb) { /* Computing MIN Computing MAX */ d__3 = 1., d__4 = abs(acoef), d__3 = max(d__3,d__4), d__4 = abs(bcoefr); d__1 = scale, d__2 = 1. / (safmin * max(d__3,d__4)); scale = min(d__1,d__2); if (lsa) { acoef = ascale * (scale * sbeta); } else { acoef = scale * acoef; } if (lsb) { bcoefr = bscale * (scale * salfar); } else { bcoefr = scale * bcoefr; } } acoefa = abs(acoef); bcoefa = abs(bcoefr); /* First component is 1 */ work[(*n << 1) + je] = 1.; xmax = 1.; /* Compute contribution from column JE of A and B to sum (See "Further Details", above.) */ i__1 = je - 1; for (jr = 1; jr <= i__1; ++jr) { work[(*n << 1) + jr] = bcoefr * b_ref(jr, je) - acoef * a_ref(jr, je); /* L260: */ } } else { /* Complex eigenvalue */ d__1 = safmin * 100.; dlag2_(&a_ref(je - 1, je - 1), lda, &b_ref(je - 1, je - 1), ldb, &d__1, &acoef, &temp, &bcoefr, &temp2, &bcoefi); if (bcoefi == 0.) { *info = je - 1; return 0; } /* Scale to avoid over/underflow */ acoefa = abs(acoef); bcoefa = abs(bcoefr) + abs(bcoefi); scale = 1.; if (acoefa * ulp < safmin && acoefa >= safmin) { scale = safmin / ulp / acoefa; } if (bcoefa * ulp < safmin && bcoefa >= safmin) { /* Computing MAX */ d__1 = scale, d__2 = safmin / ulp / bcoefa; scale = max(d__1,d__2); } if (safmin * acoefa > ascale) { scale = ascale / (safmin * acoefa); } if (safmin * bcoefa > bscale) { /* Computing MIN */ d__1 = scale, d__2 = bscale / (safmin * bcoefa); scale = min(d__1,d__2); } if (scale != 1.) { acoef = scale * acoef; acoefa = abs(acoef); bcoefr = scale * bcoefr; bcoefi = scale * bcoefi; bcoefa = abs(bcoefr) + abs(bcoefi); } /* Compute first two components of eigenvector and contribution to sums */ temp = acoef * a_ref(je, je - 1); temp2r = acoef * a_ref(je, je) - bcoefr * b_ref(je, je); temp2i = -bcoefi * b_ref(je, je); if (abs(temp) >= abs(temp2r) + abs(temp2i)) { work[(*n << 1) + je] = 1.; work[*n * 3 + je] = 0.; work[(*n << 1) + je - 1] = -temp2r / temp; work[*n * 3 + je - 1] = -temp2i / temp; } else { work[(*n << 1) + je - 1] = 1.; work[*n * 3 + je - 1] = 0.; temp = acoef * a_ref(je - 1, je); work[(*n << 1) + je] = (bcoefr * b_ref(je - 1, je - 1) - acoef * a_ref(je - 1, je - 1)) / temp; work[*n * 3 + je] = bcoefi * b_ref(je - 1, je - 1) / temp; } /* Computing MAX */ d__5 = (d__1 = work[(*n << 1) + je], abs(d__1)) + (d__2 = work[*n * 3 + je], abs(d__2)), d__6 = (d__3 = work[(* n << 1) + je - 1], abs(d__3)) + (d__4 = work[*n * 3 + je - 1], abs(d__4)); xmax = max(d__5,d__6); /* Compute contribution from columns JE and JE-1 of A and B to the sums. */ creala = acoef * work[(*n << 1) + je - 1]; cimaga = acoef * work[*n * 3 + je - 1]; crealb = bcoefr * work[(*n << 1) + je - 1] - bcoefi * work[*n * 3 + je - 1]; cimagb = bcoefi * work[(*n << 1) + je - 1] + bcoefr * work[*n * 3 + je - 1]; cre2a = acoef * work[(*n << 1) + je]; cim2a = acoef * work[*n * 3 + je]; cre2b = bcoefr * work[(*n << 1) + je] - bcoefi * work[*n * 3 + je]; cim2b = bcoefi * work[(*n << 1) + je] + bcoefr * work[*n * 3 + je]; i__1 = je - 2; for (jr = 1; jr <= i__1; ++jr) { work[(*n << 1) + jr] = -creala * a_ref(jr, je - 1) + crealb * b_ref(jr, je - 1) - cre2a * a_ref(jr, je) + cre2b * b_ref(jr, je); work[*n * 3 + jr] = -cimaga * a_ref(jr, je - 1) + cimagb * b_ref(jr, je - 1) - cim2a * a_ref(jr, je) + cim2b * b_ref(jr, je); /* L270: */ } } /* Computing MAX */ d__1 = ulp * acoefa * anorm, d__2 = ulp * bcoefa * bnorm, d__1 = max(d__1,d__2); dmin__ = max(d__1,safmin); /* Columnwise triangular solve of (a A - b B) x = 0 */ il2by2 = FALSE_; for (j = je - nw; j >= 1; --j) { /* If a 2-by-2 block, is in position j-1:j, wait until next iteration to process it (when it will be j:j+1) */ if (! il2by2 && j > 1) { if (a_ref(j, j - 1) != 0.) { il2by2 = TRUE_; goto L370; } } bdiag[0] = b_ref(j, j); if (il2by2) { na = 2; bdiag[1] = b_ref(j + 1, j + 1); } else { na = 1; } /* Compute x(j) (and x(j+1), if 2-by-2 block) */ dlaln2_(&c_false, &na, &nw, &dmin__, &acoef, &a_ref(j, j), lda, bdiag, &bdiag[1], &work[(*n << 1) + j], n, & bcoefr, &bcoefi, sum, &c__2, &scale, &temp, &iinfo); if (scale < 1.) { i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = je; for (jr = 1; jr <= i__2; ++jr) { work[(jw + 2) * *n + jr] = scale * work[(jw + 2) * *n + jr]; /* L280: */ } /* L290: */ } } /* Computing MAX */ d__1 = scale * xmax; xmax = max(d__1,temp); i__1 = nw; for (jw = 1; jw <= i__1; ++jw) { i__2 = na; for (ja = 1; ja <= i__2; ++ja) { work[(jw + 1) * *n + j + ja - 1] = sum_ref(ja, jw); /* L300: */ } /* L310: */ } /* w = w + x(j)*(a A(*,j) - b B(*,j) ) with scaling */ if (j > 1) { /* Check whether scaling is necessary for sum. */ xscale = 1. / max(1.,xmax); temp = acoefa * work[j] + bcoefa * work[*n + j]; if (il2by2) { /* Computing MAX */ d__1 = temp, d__2 = acoefa * work[j + 1] + bcoefa * work[*n + j + 1]; temp = max(d__1,d__2); } /* Computing MAX */ d__1 = max(temp,acoefa); temp = max(d__1,bcoefa); if (temp > bignum * xscale) { i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = je; for (jr = 1; jr <= i__2; ++jr) { work[(jw + 2) * *n + jr] = xscale * work[(jw + 2) * *n + jr]; /* L320: */ } /* L330: */ } xmax *= xscale; } /* Compute the contributions of the off-diagonals of column j (and j+1, if 2-by-2 block) of A and B to the sums. */ i__1 = na; for (ja = 1; ja <= i__1; ++ja) { if (ilcplx) { creala = acoef * work[(*n << 1) + j + ja - 1]; cimaga = acoef * work[*n * 3 + j + ja - 1]; crealb = bcoefr * work[(*n << 1) + j + ja - 1] - bcoefi * work[*n * 3 + j + ja - 1]; cimagb = bcoefi * work[(*n << 1) + j + ja - 1] + bcoefr * work[*n * 3 + j + ja - 1]; i__2 = j - 1; for (jr = 1; jr <= i__2; ++jr) { work[(*n << 1) + jr] = work[(*n << 1) + jr] - creala * a_ref(jr, j + ja - 1) + crealb * b_ref(jr, j + ja - 1); work[*n * 3 + jr] = work[*n * 3 + jr] - cimaga * a_ref(jr, j + ja - 1) + cimagb * b_ref(jr, j + ja - 1); /* L340: */ } } else { creala = acoef * work[(*n << 1) + j + ja - 1]; crealb = bcoefr * work[(*n << 1) + j + ja - 1]; i__2 = j - 1; for (jr = 1; jr <= i__2; ++jr) { work[(*n << 1) + jr] = work[(*n << 1) + jr] - creala * a_ref(jr, j + ja - 1) + crealb * b_ref(jr, j + ja - 1); /* L350: */ } } /* L360: */ } } il2by2 = FALSE_; L370: ; } /* Copy eigenvector to VR, back transforming if HOWMNY='B'. */ ieig -= nw; if (ilback) { i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { work[(jw + 4) * *n + jr] = work[(jw + 2) * *n + 1] * vr_ref(jr, 1); /* L380: */ } /* A series of compiler directives to defeat vectorization for the next loop */ i__2 = je; for (jc = 2; jc <= i__2; ++jc) { i__3 = *n; for (jr = 1; jr <= i__3; ++jr) { work[(jw + 4) * *n + jr] += work[(jw + 2) * *n + jc] * vr_ref(jr, jc); /* L390: */ } /* L400: */ } /* L410: */ } i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr_ref(jr, ieig + jw) = work[(jw + 4) * *n + jr]; /* L420: */ } /* L430: */ } iend = *n; } else { i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vr_ref(jr, ieig + jw) = work[(jw + 2) * *n + jr]; /* L440: */ } /* L450: */ } iend = je; } /* Scale eigenvector */ xmax = 0.; if (ilcplx) { i__1 = iend; for (j = 1; j <= i__1; ++j) { /* Computing MAX */ d__3 = xmax, d__4 = (d__1 = vr_ref(j, ieig), abs(d__1)) + (d__2 = vr_ref(j, ieig + 1), abs(d__2)); xmax = max(d__3,d__4); /* L460: */ } } else { i__1 = iend; for (j = 1; j <= i__1; ++j) { /* Computing MAX */ d__2 = xmax, d__3 = (d__1 = vr_ref(j, ieig), abs(d__1)); xmax = max(d__2,d__3); /* L470: */ } } if (xmax > safmin) { xscale = 1. / xmax; i__1 = nw - 1; for (jw = 0; jw <= i__1; ++jw) { i__2 = iend; for (jr = 1; jr <= i__2; ++jr) { vr_ref(jr, ieig + jw) = xscale * vr_ref(jr, ieig + jw) ; /* L480: */ } /* L490: */ } } L500: ; } } return 0; /* End of DTGEVC */ } /* dtgevc_ */ #undef sum_ref #undef vr_ref #undef vl_ref #undef b_ref #undef a_ref #undef sumb_ref #undef suma_ref