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DLAG2(l)) DLAG2(l)NAMEDLAG2 - compute the eigenvalues of a 2 x 2 generalized eigenvalue problem A - w B, with scaling as necessary to avoid over-/underflowSYNOPSISSUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI ) INTEGER LDA, LDB DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2 DOUBLE PRECISION A( LDA, * ), B( LDB, * )PURPOSEDLAG2 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 modi- fied 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 over- flow and, if possible, do not underflow, either.ARGUMENTSA (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION The smallest positive number s.t. 1/SAFMIN does not overflow. (This should always be DLAMCH('S')it is an argument in order to avoid having to call DLAMCH fre- quently.) SCALE1 (output) DOUBLE PRECISION A scaling factor used to avoid over-/underflow in the eigenvalue equation which defines the first eigenvalue. If the eigenvalues are complex, then the eigenval- ues 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 eigenval- ues 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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 eigenval- ues. WI (output) DOUBLE PRECISION 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-neg- ative.--LAPACK version 3.015 June 2000 DLAG2(l)

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