LAPACK

NAME¶

lag2 - lag2: 2x2 eig

SYNOPSIS¶

Functions¶

subroutine dlag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi)
DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow. subroutine slag2 (a, lda, b, ldb, safmin, scale1, scale2, wr1, wr2, wi)
SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Detailed Description¶

Function Documentation¶

subroutine dlag2 (double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision safmin, double precision scale1, double precision scale2, double precision wr1, double precision wr2, double precision wi)¶

DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.

Purpose:



 DLAG2 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.

Parameters



          A is 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



          LDA is INTEGER


          The leading dimension of the array A.  LDA >= 2.



          B is 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



          LDB is INTEGER


          The leading dimension of the array B.  LDB >= 2.

SAFMIN



          SAFMIN is 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 frequently.)

SCALE1



          SCALE1 is 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 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 threshold if the exact eigenvalue


          is sufficiently large.

SCALE2



          SCALE2 is 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


          threshold if the exact eigenvalue is sufficiently large.

WR1



          WR1 is 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



          WR2 is 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 eigenvalues.



          WI is 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-negative.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine slag2 (real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real safmin, real scale1, real scale2, real wr1, real wr2, real wi)¶

SLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.