/* dtgevc.f -- translated by f2c (version 20061008). You must link the resulting object file with libf2c: on Microsoft Windows system, link with libf2c.lib; on Linux or Unix systems, link with .../path/to/libf2c.a -lm or, if you install libf2c.a in a standard place, with -lf2c -lm -- in that order, at the end of the command line, as in cc *.o -lf2c -lm Source for libf2c is in /netlib/f2c/libf2c.zip, e.g., http://www.netlib.org/f2c/libf2c.zip */ #include "f2c.h" #include "blaswrap.h" /* Table of constant values */ static logical c_true = TRUE_; static integer c__2 = 2; static doublereal c_b34 = 1.; static integer c__1 = 1; static doublereal c_b36 = 0.; static logical c_false = FALSE_; /* Subroutine */ int dtgevc_(char *side, char *howmny, logical *select, integer *n, doublereal *s, integer *lds, doublereal *p, integer *ldp, doublereal *vl, integer *ldvl, doublereal *vr, integer *ldvr, integer *mm, integer *m, doublereal *work, integer *info) { /* System generated locals */ integer p_dim1, p_offset, s_dim1, s_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 */ integer i__, j, ja, jc, je, na, im, jr, jw, nw; doublereal big; logical lsa, lsb; doublereal ulp, sum[4] /* was [2][2] */; integer ibeg, ieig, iend; doublereal dmin__, temp, xmax, sump[4] /* was [2][2] */, sums[4] /* was [2][2] */; extern /* Subroutine */ int dlag2_(doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *); doublereal cim2a, cim2b, cre2a, cre2b, temp2, bdiag[2], acoef, scale; logical ilall; integer iside; doublereal sbeta; extern logical lsame_(char *, char *); extern /* Subroutine */ int dgemv_(char *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); logical il2by2; integer iinfo; doublereal small; logical compl; doublereal anorm, bnorm; logical compr; extern /* Subroutine */ int dlaln2_(logical *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal * , doublereal *, integer *, doublereal *, doublereal *, integer *); doublereal temp2i; extern /* Subroutine */ int dlabad_(doublereal *, doublereal *); doublereal temp2r; logical ilabad, ilbbad; doublereal acoefa, bcoefa, cimaga, cimagb; logical ilback; doublereal bcoefi, ascale, bscale, creala, crealb; extern doublereal dlamch_(char *); doublereal bcoefr, salfar, safmin; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *); doublereal xscale, bignum; extern /* Subroutine */ int xerbla_(char *, integer *); logical ilcomp, ilcplx; integer ihwmny; /* -- LAPACK routine (version 3.2) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DTGEVC computes some or all of the right and/or left eigenvectors of */ /* a pair of real matrices (S,P), where S is a quasi-triangular matrix */ /* and P is upper triangular. Matrix pairs of this type are produced by */ /* the generalized Schur factorization of a matrix pair (A,B): */ /* A = Q*S*Z**T, B = Q*P*Z**T */ /* as computed by DGGHRD + DHGEQZ. */ /* The right eigenvector x and the left eigenvector y of (S,P) */ /* corresponding to an eigenvalue w are defined by: */ /* S*x = w*P*x, (y**H)*S = w*(y**H)*P, */ /* where y**H denotes the conjugate tranpose of y. */ /* The eigenvalues are not input to this routine, but are computed */ /* directly from the diagonal blocks of S and P. */ /* This routine returns the matrices X and/or Y of right and left */ /* eigenvectors of (S,P), or the products Z*X and/or Q*Y, */ /* where Z and Q are input matrices. */ /* If Q and Z are the orthogonal factors from the generalized Schur */ /* factorization of a matrix pair (A,B), then Z*X and Q*Y */ /* are the matrices of right and left eigenvectors of (A,B). */ /* 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, */ /* backtransformed by the matrices 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 w(j) is a real eigenvalue, the corresponding */ /* real eigenvector is computed if SELECT(j) is .TRUE.. */ /* If w(j) and w(j+1) are the real and imaginary parts of a */ /* complex eigenvalue, the corresponding complex eigenvector */ /* is computed if either SELECT(j) or SELECT(j+1) is .TRUE., */ /* and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is */ /* set to .FALSE.. */ /* Not referenced if HOWMNY = 'A' or 'B'. */ /* N (input) INTEGER */ /* The order of the matrices S and P. N >= 0. */ /* S (input) DOUBLE PRECISION array, dimension (LDS,N) */ /* The upper quasi-triangular matrix S from a generalized Schur */ /* factorization, as computed by DHGEQZ. */ /* LDS (input) INTEGER */ /* The leading dimension of array S. LDS >= max(1,N). */ /* P (input) DOUBLE PRECISION array, dimension (LDP,N) */ /* The upper triangular matrix P from a generalized Schur */ /* factorization, as computed by DHGEQZ. */ /* 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks */ /* of S must be in positive diagonal form. */ /* LDP (input) INTEGER */ /* The leading dimension of array P. LDP >= 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 (S,P); */ /* if HOWMNY = 'B', the matrix Q*Y; */ /* if HOWMNY = 'S', the left eigenvectors of (S,P) specified by */ /* SELECT, stored consecutively in the columns of */ /* VL, in the same order as their eigenvalues. */ /* 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. */ /* Not referenced if SIDE = 'R'. */ /* LDVL (input) INTEGER */ /* The leading dimension of array VL. LDVL >= 1, and if */ /* SIDE = 'L' or 'B', LDVL >= N. */ /* 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 Z (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 (S,P); */ /* if HOWMNY = 'B' or 'b', the matrix Z*X; */ /* if HOWMNY = 'S' or 's', the right eigenvectors of (S,P) */ /* specified by SELECT, stored consecutively in the */ /* columns of VR, in the same order as their */ /* eigenvalues. */ /* 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. */ /* Not referenced if SIDE = 'L'. */ /* LDVR (input) INTEGER */ /* The leading dimension of the array VR. LDVR >= 1, and if */ /* SIDE = 'R' or 'B', LDVR >= N. */ /* 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 LDS (and LDP) 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. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Decode and Test the input parameters */ /* Parameter adjustments */ --select; s_dim1 = *lds; s_offset = 1 + s_dim1; s -= s_offset; p_dim1 = *ldp; p_offset = 1 + p_dim1; p -= p_offset; 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 */ 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")) { 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 (*lds < max(1,*n)) { *info = -6; } else if (*ldp < 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 (s[j + 1 + j * s_dim1] != 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 (s[j + 1 + j * s_dim1] != 0.) { if (p[j + j * p_dim1] == 0. || p[j + 1 + (j + 1) * p_dim1] == 0. || p[j + (j + 1) * p_dim1] != 0.) { ilbbad = TRUE_; } if (j < *n - 1) { if (s[j + 2 + (j + 1) * s_dim1] != 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 = s[s_dim1 + 1], abs(d__1)); if (*n > 1) { anorm += (d__1 = s[s_dim1 + 2], abs(d__1)); } bnorm = (d__1 = p[p_dim1 + 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 (s[j + (j - 1) * s_dim1] == 0.) { iend = j - 1; } else { iend = j - 2; } i__2 = iend; for (i__ = 1; i__ <= i__2; ++i__) { temp += (d__1 = s[i__ + j * s_dim1], abs(d__1)); temp2 += (d__1 = p[i__ + j * p_dim1], 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 = s[i__ + j * s_dim1], abs(d__1)); temp2 += (d__1 = p[i__ + j * p_dim1], 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 (s[je + 1 + je * s_dim1] != 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 = s[je + je * s_dim1], abs(d__1)) <= safmin && ( d__2 = p[je + je * p_dim1], abs(d__2)) <= safmin) { /* Singular matrix pencil -- return unit eigenvector */ ++ieig; i__2 = *n; for (jr = 1; jr <= i__2; ++jr) { vl[jr + ieig * vl_dim1] = 0.; /* L60: */ } vl[ieig + ieig * vl_dim1] = 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 = s[je + je * s_dim1], abs(d__1)) * ascale, d__4 = (d__2 = p[je + je * p_dim1], abs(d__2)) * bscale, d__3 = max(d__3,d__4); temp = 1. / max(d__3,safmin); salfar = temp * s[je + je * s_dim1] * ascale; sbeta = temp * p[je + je * p_dim1] * 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_(&s[je + je * s_dim1], lds, &p[je + je * p_dim1], ldp, & 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 * s[je + 1 + je * s_dim1]; temp2r = acoef * s[je + je * s_dim1] - bcoefr * p[je + je * p_dim1]; temp2i = -bcoefi * p[je + je * p_dim1]; 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 * s[je + (je + 1) * s_dim1]; work[(*n << 1) + je] = (bcoefr * p[je + 1 + (je + 1) * p_dim1] - acoef * s[je + 1 + (je + 1) * s_dim1]) / temp; work[*n * 3 + je] = bcoefi * p[je + 1 + (je + 1) * p_dim1] / 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] = p[j + j * p_dim1]; if (j < *n) { if (s[j + 1 + j * s_dim1] != 0.) { il2by2 = TRUE_; bdiag[1] = p[j + 1 + (j + 1) * p_dim1]; 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*S(k,j) - b*P(k,j) )*x(k) */ /* k=je */ /* To reduce the op count, this is done as */ /* _ j-1 _ j-1 */ /* a*conjg( sum S(k,j)*x(k) ) - b*conjg( sum P(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) { sums[ja + (jw << 1) - 3] = 0.; sump[ja + (jw << 1) - 3] = 0.; i__5 = j - 1; for (jr = je; jr <= i__5; ++jr) { sums[ja + (jw << 1) - 3] += s[jr + (j + ja - 1) * s_dim1] * work[(jw + 1) * *n + jr]; sump[ja + (jw << 1) - 3] += p[jr + (j + ja - 1) * p_dim1] * 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[ja - 1] = -acoef * sums[ja - 1] + bcoefr * sump[ ja - 1] - bcoefi * sump[ja + 1]; sum[ja + 1] = -acoef * sums[ja + 1] + bcoefr * sump[ ja + 1] + bcoefi * sump[ja - 1]; } else { sum[ja - 1] = -acoef * sums[ja - 1] + bcoefr * sump[ 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, &s[j + j * s_dim1] , lds, 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_b34, &vl[je * vl_dim1 + 1], ldvl, &work[(jw + 2) * *n + je], &c__1, &c_b36, &work[( jw + 4) * *n + 1], &c__1); /* L170: */ } dlacpy_(" ", n, &nw, &work[(*n << 2) + 1], n, &vl[je * vl_dim1 + 1], ldvl); ibeg = 1; } else { dlacpy_(" ", n, &nw, &work[(*n << 1) + 1], n, &vl[ieig * vl_dim1 + 1], 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[j + ieig * vl_dim1], abs( d__1)) + (d__2 = vl[j + (ieig + 1) * vl_dim1], 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[j + ieig * vl_dim1], 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[jr + (ieig + jw) * vl_dim1] = xscale * vl[jr + ( ieig + jw) * vl_dim1]; /* 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 (s[je + (je - 1) * s_dim1] != 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 = s[je + je * s_dim1], abs(d__1)) <= safmin && ( d__2 = p[je + je * p_dim1], abs(d__2)) <= safmin) { /* Singular matrix pencil -- unit eigenvector */ --ieig; i__1 = *n; for (jr = 1; jr <= i__1; ++jr) { vr[jr + ieig * vr_dim1] = 0.; /* L230: */ } vr[ieig + ieig * vr_dim1] = 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 = s[je + je * s_dim1], abs(d__1)) * ascale, d__4 = (d__2 = p[je + je * p_dim1], abs(d__2)) * bscale, d__3 = max(d__3,d__4); temp = 1. / max(d__3,safmin); salfar = temp * s[je + je * s_dim1] * ascale; sbeta = temp * p[je + je * p_dim1] * 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 * p[jr + je * p_dim1] - acoef * s[jr + je * s_dim1]; /* L260: */ } } else { /* Complex eigenvalue */ d__1 = safmin * 100.; dlag2_(&s[je - 1 + (je - 1) * s_dim1], lds, &p[je - 1 + (je - 1) * p_dim1], ldp, &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 * s[je + (je - 1) * s_dim1]; temp2r = acoef * s[je + je * s_dim1] - bcoefr * p[je + je * p_dim1]; temp2i = -bcoefi * p[je + je * p_dim1]; 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 * s[je - 1 + je * s_dim1]; work[(*n << 1) + je] = (bcoefr * p[je - 1 + (je - 1) * p_dim1] - acoef * s[je - 1 + (je - 1) * s_dim1]) / temp; work[*n * 3 + je] = bcoefi * p[je - 1 + (je - 1) * p_dim1] / 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 * s[jr + (je - 1) * s_dim1] + crealb * p[jr + (je - 1) * p_dim1] - cre2a * s[ jr + je * s_dim1] + cre2b * p[jr + je * p_dim1]; work[*n * 3 + jr] = -cimaga * s[jr + (je - 1) * s_dim1] + cimagb * p[jr + (je - 1) * p_dim1] - cim2a * s[jr + je * s_dim1] + cim2b * p[jr + je * p_dim1]; /* 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 (s[j + (j - 1) * s_dim1] != 0.) { il2by2 = TRUE_; goto L370; } } bdiag[0] = p[j + j * p_dim1]; if (il2by2) { na = 2; bdiag[1] = p[j + 1 + (j + 1) * p_dim1]; } else { na = 1; } /* Compute x(j) (and x(j+1), if 2-by-2 block) */ dlaln2_(&c_false, &na, &nw, &dmin__, &acoef, &s[j + j * s_dim1], lds, 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[ja + (jw << 1) - 3]; /* L300: */ } /* L310: */ } /* w = w + x(j)*(a S(*,j) - b P(*,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 * s[jr + (j + ja - 1) * s_dim1] + crealb * p[jr + (j + ja - 1) * p_dim1]; work[*n * 3 + jr] = work[*n * 3 + jr] - cimaga * s[jr + (j + ja - 1) * s_dim1] + cimagb * p[jr + (j + ja - 1) * p_dim1]; /* 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 * s[jr + (j + ja - 1) * s_dim1] + crealb * p[jr + (j + ja - 1) * p_dim1]; /* 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[jr + vr_dim1]; /* 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[jr + jc * vr_dim1]; /* 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[jr + (ieig + jw) * vr_dim1] = 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[jr + (ieig + jw) * vr_dim1] = 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[j + ieig * vr_dim1], abs( d__1)) + (d__2 = vr[j + (ieig + 1) * vr_dim1], 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[j + ieig * vr_dim1], 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[jr + (ieig + jw) * vr_dim1] = xscale * vr[jr + ( ieig + jw) * vr_dim1]; /* L480: */ } /* L490: */ } } L500: ; } } return 0; /* End of DTGEVC */ } /* dtgevc_ */