Mercurial > octave

annotate libcruft/lapack/slag2.f @ 7789:82be108cc558

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
First attempt at single precision tyeps * * * corrections to qrupdate single precision routines * * * prefer demotion to single over promotion to double * * * Add single precision support to log2 function * * * Trivial PROJECT file update * * * Cache optimized hermitian/transpose methods * * * Add tests for tranpose/hermitian and ChangeLog entry for new transpose code
author David Bateman <dbateman@free.fr>
date Sun, 27 Apr 2008 22:34:17 +0200
parents
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
7789
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
1 SUBROUTINE SLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
2 $ WR2, WI )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
3 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
4 * -- LAPACK auxiliary routine (version 3.1) --
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
5 * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
6 * November 2006
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
7 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
8 * .. Scalar Arguments ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
9 INTEGER LDA, LDB
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
10 REAL SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
11 * ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
12 * .. Array Arguments ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
13 REAL A( LDA, * ), B( LDB, * )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
14 * ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
15 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
16 * Purpose
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
17 * =======
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
18 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
19 * SLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
20 * problem A - w B, with scaling as necessary to avoid over-/underflow.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
21 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
22 * The scaling factor "s" results in a modified eigenvalue equation
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
23 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
24 * s A - w B
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
25 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
26 * where s is a non-negative scaling factor chosen so that w, w B,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
27 * and s A do not overflow and, if possible, do not underflow, either.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
28 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
29 * Arguments
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
30 * =========
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
31 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
32 * A (input) REAL array, dimension (LDA, 2)
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
33 * On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
34 * is less than 1/SAFMIN. Entries less than
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
35 * sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
36 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
37 * LDA (input) INTEGER
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
38 * The leading dimension of the array A. LDA >= 2.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
39 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
40 * B (input) REAL array, dimension (LDB, 2)
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
41 * On entry, the 2 x 2 upper triangular matrix B. It is
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
42 * assumed that the one-norm of B is less than 1/SAFMIN. The
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
43 * diagonals should be at least sqrt(SAFMIN) times the largest
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
44 * element of B (in absolute value); if a diagonal is smaller
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
45 * than that, then +/- sqrt(SAFMIN) will be used instead of
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
46 * that diagonal.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
47 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
48 * LDB (input) INTEGER
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
49 * The leading dimension of the array B. LDB >= 2.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
50 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
51 * SAFMIN (input) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
52 * The smallest positive number s.t. 1/SAFMIN does not
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
53 * overflow. (This should always be SLAMCH('S') -- it is an
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
54 * argument in order to avoid having to call SLAMCH frequently.)
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
55 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
56 * SCALE1 (output) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
57 * A scaling factor used to avoid over-/underflow in the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
58 * eigenvalue equation which defines the first eigenvalue. If
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
59 * the eigenvalues are complex, then the eigenvalues are
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
60 * ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
61 * exponent range of the machine), SCALE1=SCALE2, and SCALE1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
62 * will always be positive. If the eigenvalues are real, then
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
63 * the first (real) eigenvalue is WR1 / SCALE1 , but this may
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
64 * overflow or underflow, and in fact, SCALE1 may be zero or
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
65 * less than the underflow threshhold if the exact eigenvalue
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
66 * is sufficiently large.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
67 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
68 * SCALE2 (output) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
69 * A scaling factor used to avoid over-/underflow in the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
70 * eigenvalue equation which defines the second eigenvalue. If
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
71 * the eigenvalues are complex, then SCALE2=SCALE1. If the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
72 * eigenvalues are real, then the second (real) eigenvalue is
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
73 * WR2 / SCALE2 , but this may overflow or underflow, and in
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
74 * fact, SCALE2 may be zero or less than the underflow
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
75 * threshhold if the exact eigenvalue is sufficiently large.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
76 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
77 * WR1 (output) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
78 * If the eigenvalue is real, then WR1 is SCALE1 times the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
79 * eigenvalue closest to the (2,2) element of A B**(-1). If the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
80 * eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
81 * part of the eigenvalues.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
82 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
83 * WR2 (output) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
84 * If the eigenvalue is real, then WR2 is SCALE2 times the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
85 * other eigenvalue. If the eigenvalue is complex, then
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
86 * WR1=WR2 is SCALE1 times the real part of the eigenvalues.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
87 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
88 * WI (output) REAL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
89 * If the eigenvalue is real, then WI is zero. If the
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
90 * eigenvalue is complex, then WI is SCALE1 times the imaginary
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
91 * part of the eigenvalues. WI will always be non-negative.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
92 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
93 * =====================================================================
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
94 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
95 * .. Parameters ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
96 REAL ZERO, ONE, TWO
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
97 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
98 REAL HALF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
99 PARAMETER ( HALF = ONE / TWO )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
100 REAL FUZZY1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
101 PARAMETER ( FUZZY1 = ONE+1.0E-5 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
102 * ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
103 * .. Local Scalars ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
104 REAL A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
105 $ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
106 $ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
107 $ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
108 $ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
109 $ WSCALE, WSIZE, WSMALL
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
110 * ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
111 * .. Intrinsic Functions ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
112 INTRINSIC ABS, MAX, MIN, SIGN, SQRT
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
113 * ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
114 * .. Executable Statements ..
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
115 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
116 RTMIN = SQRT( SAFMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
117 RTMAX = ONE / RTMIN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
118 SAFMAX = ONE / SAFMIN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
119 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
120 * Scale A
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
121 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
122 ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
123 $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
124 ASCALE = ONE / ANORM
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
125 A11 = ASCALE*A( 1, 1 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
126 A21 = ASCALE*A( 2, 1 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
127 A12 = ASCALE*A( 1, 2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
128 A22 = ASCALE*A( 2, 2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
129 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
130 * Perturb B if necessary to insure non-singularity
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
131 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
132 B11 = B( 1, 1 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
133 B12 = B( 1, 2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
134 B22 = B( 2, 2 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
135 BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
136 IF( ABS( B11 ).LT.BMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
137 $ B11 = SIGN( BMIN, B11 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
138 IF( ABS( B22 ).LT.BMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
139 $ B22 = SIGN( BMIN, B22 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
140 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
141 * Scale B
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
142 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
143 BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
144 BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
145 BSCALE = ONE / BSIZE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
146 B11 = B11*BSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
147 B12 = B12*BSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
148 B22 = B22*BSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
149 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
150 * Compute larger eigenvalue by method described by C. van Loan
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
151 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
152 * ( AS is A shifted by -SHIFT*B )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
153 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
154 BINV11 = ONE / B11
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
155 BINV22 = ONE / B22
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
156 S1 = A11*BINV11
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
157 S2 = A22*BINV22
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
158 IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
159 AS12 = A12 - S1*B12
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
160 AS22 = A22 - S1*B22
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
161 SS = A21*( BINV11*BINV22 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
162 ABI22 = AS22*BINV22 - SS*B12
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
163 PP = HALF*ABI22
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
164 SHIFT = S1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
165 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
166 AS12 = A12 - S2*B12
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
167 AS11 = A11 - S2*B11
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
168 SS = A21*( BINV11*BINV22 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
169 ABI22 = -SS*B12
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
170 PP = HALF*( AS11*BINV11+ABI22 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
171 SHIFT = S2
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
172 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
173 QQ = SS*AS12
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
174 IF( ABS( PP*RTMIN ).GE.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
175 DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
176 R = SQRT( ABS( DISCR ) )*RTMAX
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
177 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
178 IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
179 DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
180 R = SQRT( ABS( DISCR ) )*RTMIN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
181 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
182 DISCR = PP**2 + QQ
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
183 R = SQRT( ABS( DISCR ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
184 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
185 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
186 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
187 * Note: the test of R in the following IF is to cover the case when
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
188 * DISCR is small and negative and is flushed to zero during
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
189 * the calculation of R. On machines which have a consistent
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
190 * flush-to-zero threshhold and handle numbers above that
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
191 * threshhold correctly, it would not be necessary.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
192 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
193 IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
194 SUM = PP + SIGN( R, PP )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
195 DIFF = PP - SIGN( R, PP )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
196 WBIG = SHIFT + SUM
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
197 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
198 * Compute smaller eigenvalue
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
199 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
200 WSMALL = SHIFT + DIFF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
201 IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
202 WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
203 WSMALL = WDET / WBIG
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
204 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
205 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
206 * Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
207 * for WR1.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
208 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
209 IF( PP.GT.ABI22 ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
210 WR1 = MIN( WBIG, WSMALL )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
211 WR2 = MAX( WBIG, WSMALL )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
212 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
213 WR1 = MAX( WBIG, WSMALL )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
214 WR2 = MIN( WBIG, WSMALL )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
215 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
216 WI = ZERO
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
217 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
218 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
219 * Complex eigenvalues
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
220 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
221 WR1 = SHIFT + PP
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
222 WR2 = WR1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
223 WI = R
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
224 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
225 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
226 * Further scaling to avoid underflow and overflow in computing
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
227 * SCALE1 and overflow in computing w*B.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
228 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
229 * This scale factor (WSCALE) is bounded from above using C1 and C2,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
230 * and from below using C3 and C4.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
231 * C1 implements the condition s A must never overflow.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
232 * C2 implements the condition w B must never overflow.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
233 * C3, with C2,
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
234 * implement the condition that s A - w B must never overflow.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
235 * C4 implements the condition s should not underflow.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
236 * C5 implements the condition max(s,|w|) should be at least 2.
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
237 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
238 C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
239 C2 = SAFMIN*MAX( ONE, BNORM )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
240 C3 = BSIZE*SAFMIN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
241 IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
242 C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
243 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
244 C4 = ONE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
245 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
246 IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
247 C5 = MIN( ONE, ASCALE*BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
248 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
249 C5 = ONE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
250 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
251 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
252 * Scale first eigenvalue
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
253 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
254 WABS = ABS( WR1 ) + ABS( WI )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
255 WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
256 $ MIN( C4, HALF*MAX( WABS, C5 ) ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
257 IF( WSIZE.NE.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
258 WSCALE = ONE / WSIZE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
259 IF( WSIZE.GT.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
260 SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
261 $ MIN( ASCALE, BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
262 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
263 SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
264 $ MAX( ASCALE, BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
265 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
266 WR1 = WR1*WSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
267 IF( WI.NE.ZERO ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
268 WI = WI*WSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
269 WR2 = WR1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
270 SCALE2 = SCALE1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
271 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
272 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
273 SCALE1 = ASCALE*BSIZE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
274 SCALE2 = SCALE1
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
275 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
276 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
277 * Scale second eigenvalue (if real)
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
278 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
279 IF( WI.EQ.ZERO ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
280 WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
281 $ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
282 IF( WSIZE.NE.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
283 WSCALE = ONE / WSIZE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
284 IF( WSIZE.GT.ONE ) THEN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
285 SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
286 $ MIN( ASCALE, BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
287 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
288 SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
289 $ MAX( ASCALE, BSIZE )
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
290 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
291 WR2 = WR2*WSCALE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
292 ELSE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
293 SCALE2 = ASCALE*BSIZE
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
294 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
295 END IF
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
296 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
297 * End of SLAG2
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
298 *
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
299 RETURN
82be108cc558 First attempt at single precision tyeps
David Bateman <dbateman@free.fr>
parents:
diff changeset
300 END