#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int slag2_(real *a, integer *lda, real *b, integer *ldb, real *safmin, real *scale1, real *scale2, real *wr1, real *wr2, real * wi) { /* -- LAPACK auxiliary routine (version 3.1) -- Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. November 2006 Purpose ======= SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflow. The scaling factor "s" results in a modified eigenvalue equation s A - w B where s is a non-negative scaling factor chosen so that w, w B, and s A do not overflow and, if possible, do not underflow, either. Arguments ========= A (input) REAL array, dimension (LDA, 2) On entry, the 2 x 2 matrix A. It is assumed that its 1-norm is less than 1/SAFMIN. Entries less than sqrt(SAFMIN)*norm(A) are subject to being treated as zero. LDA (input) INTEGER The leading dimension of the array A. LDA >= 2. B (input) REAL array, dimension (LDB, 2) On entry, the 2 x 2 upper triangular matrix B. It is assumed that the one-norm of B is less than 1/SAFMIN. The diagonals should be at least sqrt(SAFMIN) times the largest element of B (in absolute value); if a diagonal is smaller than that, then +/- sqrt(SAFMIN) will be used instead of that diagonal. LDB (input) INTEGER The leading dimension of the array B. LDB >= 2. SAFMIN (input) REAL The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be SLAMCH('S') -- it is an argument in order to avoid having to call SLAMCH frequently.) SCALE1 (output) REAL A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenvalues are ( WR1 +/- WI i ) / SCALE1 (which may lie outside the exponent range of the machine), SCALE1=SCALE2, and SCALE1 will always be positive. If the eigenvalues are real, then the first (real) eigenvalue is WR1 / SCALE1 , but this may overflow or underflow, and in fact, SCALE1 may be zero or less than the underflow threshhold if the exact eigenvalue is sufficiently large. SCALE2 (output) REAL A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the second eigenvalue. If the eigenvalues are complex, then SCALE2=SCALE1. If the eigenvalues are real, then the second (real) eigenvalue is WR2 / SCALE2 , but this may overflow or underflow, and in fact, SCALE2 may be zero or less than the underflow threshhold if the exact eigenvalue is sufficiently large. WR1 (output) REAL If the eigenvalue is real, then WR1 is SCALE1 times the eigenvalue closest to the (2,2) element of A B**(-1). If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues. WR2 (output) REAL If the eigenvalue is real, then WR2 is SCALE2 times the other eigenvalue. If the eigenvalue is complex, then WR1=WR2 is SCALE1 times the real part of the eigenvalues. WI (output) REAL If the eigenvalue is real, then WI is zero. If the eigenvalue is complex, then WI is SCALE1 times the imaginary part of the eigenvalues. WI will always be non-negative. ===================================================================== Parameter adjustments */ /* System generated locals */ integer a_dim1, a_offset, b_dim1, b_offset; real r__1, r__2, r__3, r__4, r__5, r__6; /* Builtin functions */ double sqrt(doublereal), r_sign(real *, real *); /* Local variables */ static real r__, c1, c2, c3, c4, c5, s1, s2, a11, a12, a21, a22, b11, b12, b22, pp, qq, ss, as11, as12, as22, sum, abi22, diff, bmin, wbig, wabs, wdet, binv11, binv22, discr, anorm, bnorm, bsize, shift, rtmin, rtmax, wsize, ascale, bscale, wscale, safmax, wsmall; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; /* Function Body */ rtmin = sqrt(*safmin); rtmax = 1.f / rtmin; safmax = 1.f / *safmin; /* Scale A Computing MAX */ r__5 = (r__1 = a[a_dim1 + 1], dabs(r__1)) + (r__2 = a[a_dim1 + 2], dabs( r__2)), r__6 = (r__3 = a[(a_dim1 << 1) + 1], dabs(r__3)) + (r__4 = a[(a_dim1 << 1) + 2], dabs(r__4)), r__5 = max(r__5,r__6); anorm = dmax(r__5,*safmin); ascale = 1.f / anorm; a11 = ascale * a[a_dim1 + 1]; a21 = ascale * a[a_dim1 + 2]; a12 = ascale * a[(a_dim1 << 1) + 1]; a22 = ascale * a[(a_dim1 << 1) + 2]; /* Perturb B if necessary to insure non-singularity */ b11 = b[b_dim1 + 1]; b12 = b[(b_dim1 << 1) + 1]; b22 = b[(b_dim1 << 1) + 2]; /* Computing MAX */ r__1 = dabs(b11), r__2 = dabs(b12), r__1 = max(r__1,r__2), r__2 = dabs( b22), r__1 = max(r__1,r__2); bmin = rtmin * dmax(r__1,rtmin); if (dabs(b11) < bmin) { b11 = r_sign(&bmin, &b11); } if (dabs(b22) < bmin) { b22 = r_sign(&bmin, &b22); } /* Scale B Computing MAX */ r__1 = dabs(b11), r__2 = dabs(b12) + dabs(b22), r__1 = max(r__1,r__2); bnorm = dmax(r__1,*safmin); /* Computing MAX */ r__1 = dabs(b11), r__2 = dabs(b22); bsize = dmax(r__1,r__2); bscale = 1.f / bsize; b11 *= bscale; b12 *= bscale; b22 *= bscale; /* Compute larger eigenvalue by method described by C. van Loan ( AS is A shifted by -SHIFT*B ) */ binv11 = 1.f / b11; binv22 = 1.f / b22; s1 = a11 * binv11; s2 = a22 * binv22; if (dabs(s1) <= dabs(s2)) { as12 = a12 - s1 * b12; as22 = a22 - s1 * b22; ss = a21 * (binv11 * binv22); abi22 = as22 * binv22 - ss * b12; pp = abi22 * .5f; shift = s1; } else { as12 = a12 - s2 * b12; as11 = a11 - s2 * b11; ss = a21 * (binv11 * binv22); abi22 = -ss * b12; pp = (as11 * binv11 + abi22) * .5f; shift = s2; } qq = ss * as12; if ((r__1 = pp * rtmin, dabs(r__1)) >= 1.f) { /* Computing 2nd power */ r__1 = rtmin * pp; discr = r__1 * r__1 + qq * *safmin; r__ = sqrt((dabs(discr))) * rtmax; } else { /* Computing 2nd power */ r__1 = pp; if (r__1 * r__1 + dabs(qq) <= *safmin) { /* Computing 2nd power */ r__1 = rtmax * pp; discr = r__1 * r__1 + qq * safmax; r__ = sqrt((dabs(discr))) * rtmin; } else { /* Computing 2nd power */ r__1 = pp; discr = r__1 * r__1 + qq; r__ = sqrt((dabs(discr))); } } /* Note: the test of R in the following IF is to cover the case when DISCR is small and negative and is flushed to zero during the calculation of R. On machines which have a consistent flush-to-zero threshhold and handle numbers above that threshhold correctly, it would not be necessary. */ if (discr >= 0.f || r__ == 0.f) { sum = pp + r_sign(&r__, &pp); diff = pp - r_sign(&r__, &pp); wbig = shift + sum; /* Compute smaller eigenvalue */ wsmall = shift + diff; /* Computing MAX */ r__1 = dabs(wsmall); if (dabs(wbig) * .5f > dmax(r__1,*safmin)) { wdet = (a11 * a22 - a12 * a21) * (binv11 * binv22); wsmall = wdet / wbig; } /* Choose (real) eigenvalue closest to 2,2 element of A*B**(-1) for WR1. */ if (pp > abi22) { *wr1 = dmin(wbig,wsmall); *wr2 = dmax(wbig,wsmall); } else { *wr1 = dmax(wbig,wsmall); *wr2 = dmin(wbig,wsmall); } *wi = 0.f; } else { /* Complex eigenvalues */ *wr1 = shift + pp; *wr2 = *wr1; *wi = r__; } /* Further scaling to avoid underflow and overflow in computing SCALE1 and overflow in computing w*B. This scale factor (WSCALE) is bounded from above using C1 and C2, and from below using C3 and C4. C1 implements the condition s A must never overflow. C2 implements the condition w B must never overflow. C3, with C2, implement the condition that s A - w B must never overflow. C4 implements the condition s should not underflow. C5 implements the condition max(s,|w|) should be at least 2. */ c1 = bsize * (*safmin * dmax(1.f,ascale)); c2 = *safmin * dmax(1.f,bnorm); c3 = bsize * *safmin; if (ascale <= 1.f && bsize <= 1.f) { /* Computing MIN */ r__1 = 1.f, r__2 = ascale / *safmin * bsize; c4 = dmin(r__1,r__2); } else { c4 = 1.f; } if (ascale <= 1.f || bsize <= 1.f) { /* Computing MIN */ r__1 = 1.f, r__2 = ascale * bsize; c5 = dmin(r__1,r__2); } else { c5 = 1.f; } /* Scale first eigenvalue */ wabs = dabs(*wr1) + dabs(*wi); /* Computing MAX Computing MIN */ r__3 = c4, r__4 = dmax(wabs,c5) * .5f; r__1 = max(*safmin,c1), r__2 = (wabs * c2 + c3) * 1.0000100000000001f, r__1 = max(r__1,r__2), r__2 = dmin(r__3,r__4); wsize = dmax(r__1,r__2); if (wsize != 1.f) { wscale = 1.f / wsize; if (wsize > 1.f) { *scale1 = dmax(ascale,bsize) * wscale * dmin(ascale,bsize); } else { *scale1 = dmin(ascale,bsize) * wscale * dmax(ascale,bsize); } *wr1 *= wscale; if (*wi != 0.f) { *wi *= wscale; *wr2 = *wr1; *scale2 = *scale1; } } else { *scale1 = ascale * bsize; *scale2 = *scale1; } /* Scale second eigenvalue (if real) */ if (*wi == 0.f) { /* Computing MAX Computing MIN Computing MAX */ r__5 = dabs(*wr2); r__3 = c4, r__4 = dmax(r__5,c5) * .5f; r__1 = max(*safmin,c1), r__2 = (dabs(*wr2) * c2 + c3) * 1.0000100000000001f, r__1 = max(r__1,r__2), r__2 = dmin(r__3, r__4); wsize = dmax(r__1,r__2); if (wsize != 1.f) { wscale = 1.f / wsize; if (wsize > 1.f) { *scale2 = dmax(ascale,bsize) * wscale * dmin(ascale,bsize); } else { *scale2 = dmin(ascale,bsize) * wscale * dmax(ascale,bsize); } *wr2 *= wscale; } else { *scale2 = ascale * bsize; } } /* End of SLAG2 */ return 0; } /* slag2_ */