DLAG2(3F) DLAG2(3F)
DLAG2 - compute the eigenvalues of a 2 x 2 generalized eigenvalue problem
A - w B, with scaling as necessary to avoid over-/underflow
SUBROUTINE 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, * )
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.
A (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 frequently.)
SCALE1 (output) DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the eigenvalue
equation which defines the first eigenvalue. If the eigenvalues
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DLAG2(3F) DLAG2(3F)
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) 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 eigenvalues.
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-negative.
DLAG2(3F) DLAG2(3F)
DLAG2 - compute the eigenvalues of a 2 x 2 generalized eigenvalue problem
A - w B, with scaling as necessary to avoid over-/underflow
SUBROUTINE 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, * )
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.
A (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 frequently.)
SCALE1 (output) DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the eigenvalue
equation which defines the first eigenvalue. If the eigenvalues
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DLAG2(3F) DLAG2(3F)
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) 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 eigenvalues.
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-negative.
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