1: SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
2: $ WR2, WI )
3: *
4: * -- LAPACK auxiliary routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: INTEGER LDA, LDB
11: DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
21: * problem A - w B, with scaling as necessary to avoid over-/underflow.
22: *
23: * The scaling factor "s" results in a modified eigenvalue equation
24: *
25: * s A - w B
26: *
27: * where s is a non-negative scaling factor chosen so that w, w B,
28: * and s A do not overflow and, if possible, do not underflow, either.
29: *
30: * Arguments
31: * =========
32: *
33: * A (input) DOUBLE PRECISION array, dimension (LDA, 2)
34: * On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
35: * is less than 1/SAFMIN. Entries less than
36: * sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
37: *
38: * LDA (input) INTEGER
39: * The leading dimension of the array A. LDA >= 2.
40: *
41: * B (input) DOUBLE PRECISION array, dimension (LDB, 2)
42: * On entry, the 2 x 2 upper triangular matrix B. It is
43: * assumed that the one-norm of B is less than 1/SAFMIN. The
44: * diagonals should be at least sqrt(SAFMIN) times the largest
45: * element of B (in absolute value); if a diagonal is smaller
46: * than that, then +/- sqrt(SAFMIN) will be used instead of
47: * that diagonal.
48: *
49: * LDB (input) INTEGER
50: * The leading dimension of the array B. LDB >= 2.
51: *
52: * SAFMIN (input) DOUBLE PRECISION
53: * The smallest positive number s.t. 1/SAFMIN does not
54: * overflow. (This should always be DLAMCH('S') -- it is an
55: * argument in order to avoid having to call DLAMCH frequently.)
56: *
57: * SCALE1 (output) DOUBLE PRECISION
58: * A scaling factor used to avoid over-/underflow in the
59: * eigenvalue equation which defines the first eigenvalue. If
60: * the eigenvalues are complex, then the eigenvalues are
61: * ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
62: * exponent range of the machine), SCALE1=SCALE2, and SCALE1
63: * will always be positive. If the eigenvalues are real, then
64: * the first (real) eigenvalue is WR1 / SCALE1 , but this may
65: * overflow or underflow, and in fact, SCALE1 may be zero or
66: * less than the underflow threshhold if the exact eigenvalue
67: * is sufficiently large.
68: *
69: * SCALE2 (output) DOUBLE PRECISION
70: * A scaling factor used to avoid over-/underflow in the
71: * eigenvalue equation which defines the second eigenvalue. If
72: * the eigenvalues are complex, then SCALE2=SCALE1. If the
73: * eigenvalues are real, then the second (real) eigenvalue is
74: * WR2 / SCALE2 , but this may overflow or underflow, and in
75: * fact, SCALE2 may be zero or less than the underflow
76: * threshhold if the exact eigenvalue is sufficiently large.
77: *
78: * WR1 (output) DOUBLE PRECISION
79: * If the eigenvalue is real, then WR1 is SCALE1 times the
80: * eigenvalue closest to the (2,2) element of A B**(-1). If the
81: * eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
82: * part of the eigenvalues.
83: *
84: * WR2 (output) DOUBLE PRECISION
85: * If the eigenvalue is real, then WR2 is SCALE2 times the
86: * other eigenvalue. If the eigenvalue is complex, then
87: * WR1=WR2 is SCALE1 times the real part of the eigenvalues.
88: *
89: * WI (output) DOUBLE PRECISION
90: * If the eigenvalue is real, then WI is zero. If the
91: * eigenvalue is complex, then WI is SCALE1 times the imaginary
92: * part of the eigenvalues. WI will always be non-negative.
93: *
94: * =====================================================================
95: *
96: * .. Parameters ..
97: DOUBLE PRECISION ZERO, ONE, TWO
98: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
99: DOUBLE PRECISION HALF
100: PARAMETER ( HALF = ONE / TWO )
101: DOUBLE PRECISION FUZZY1
102: PARAMETER ( FUZZY1 = ONE+1.0D-5 )
103: * ..
104: * .. Local Scalars ..
105: DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
106: $ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
107: $ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
108: $ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
109: $ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
110: $ WSCALE, WSIZE, WSMALL
111: * ..
112: * .. Intrinsic Functions ..
113: INTRINSIC ABS, MAX, MIN, SIGN, SQRT
114: * ..
115: * .. Executable Statements ..
116: *
117: RTMIN = SQRT( SAFMIN )
118: RTMAX = ONE / RTMIN
119: SAFMAX = ONE / SAFMIN
120: *
121: * Scale A
122: *
123: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
124: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
125: ASCALE = ONE / ANORM
126: A11 = ASCALE*A( 1, 1 )
127: A21 = ASCALE*A( 2, 1 )
128: A12 = ASCALE*A( 1, 2 )
129: A22 = ASCALE*A( 2, 2 )
130: *
131: * Perturb B if necessary to insure non-singularity
132: *
133: B11 = B( 1, 1 )
134: B12 = B( 1, 2 )
135: B22 = B( 2, 2 )
136: BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
137: IF( ABS( B11 ).LT.BMIN )
138: $ B11 = SIGN( BMIN, B11 )
139: IF( ABS( B22 ).LT.BMIN )
140: $ B22 = SIGN( BMIN, B22 )
141: *
142: * Scale B
143: *
144: BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
145: BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
146: BSCALE = ONE / BSIZE
147: B11 = B11*BSCALE
148: B12 = B12*BSCALE
149: B22 = B22*BSCALE
150: *
151: * Compute larger eigenvalue by method described by C. van Loan
152: *
153: * ( AS is A shifted by -SHIFT*B )
154: *
155: BINV11 = ONE / B11
156: BINV22 = ONE / B22
157: S1 = A11*BINV11
158: S2 = A22*BINV22
159: IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
160: AS12 = A12 - S1*B12
161: AS22 = A22 - S1*B22
162: SS = A21*( BINV11*BINV22 )
163: ABI22 = AS22*BINV22 - SS*B12
164: PP = HALF*ABI22
165: SHIFT = S1
166: ELSE
167: AS12 = A12 - S2*B12
168: AS11 = A11 - S2*B11
169: SS = A21*( BINV11*BINV22 )
170: ABI22 = -SS*B12
171: PP = HALF*( AS11*BINV11+ABI22 )
172: SHIFT = S2
173: END IF
174: QQ = SS*AS12
175: IF( ABS( PP*RTMIN ).GE.ONE ) THEN
176: DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
177: R = SQRT( ABS( DISCR ) )*RTMAX
178: ELSE
179: IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
180: DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
181: R = SQRT( ABS( DISCR ) )*RTMIN
182: ELSE
183: DISCR = PP**2 + QQ
184: R = SQRT( ABS( DISCR ) )
185: END IF
186: END IF
187: *
188: * Note: the test of R in the following IF is to cover the case when
189: * DISCR is small and negative and is flushed to zero during
190: * the calculation of R. On machines which have a consistent
191: * flush-to-zero threshhold and handle numbers above that
192: * threshhold correctly, it would not be necessary.
193: *
194: IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
195: SUM = PP + SIGN( R, PP )
196: DIFF = PP - SIGN( R, PP )
197: WBIG = SHIFT + SUM
198: *
199: * Compute smaller eigenvalue
200: *
201: WSMALL = SHIFT + DIFF
202: IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
203: WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
204: WSMALL = WDET / WBIG
205: END IF
206: *
207: * Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
208: * for WR1.
209: *
210: IF( PP.GT.ABI22 ) THEN
211: WR1 = MIN( WBIG, WSMALL )
212: WR2 = MAX( WBIG, WSMALL )
213: ELSE
214: WR1 = MAX( WBIG, WSMALL )
215: WR2 = MIN( WBIG, WSMALL )
216: END IF
217: WI = ZERO
218: ELSE
219: *
220: * Complex eigenvalues
221: *
222: WR1 = SHIFT + PP
223: WR2 = WR1
224: WI = R
225: END IF
226: *
227: * Further scaling to avoid underflow and overflow in computing
228: * SCALE1 and overflow in computing w*B.
229: *
230: * This scale factor (WSCALE) is bounded from above using C1 and C2,
231: * and from below using C3 and C4.
232: * C1 implements the condition s A must never overflow.
233: * C2 implements the condition w B must never overflow.
234: * C3, with C2,
235: * implement the condition that s A - w B must never overflow.
236: * C4 implements the condition s should not underflow.
237: * C5 implements the condition max(s,|w|) should be at least 2.
238: *
239: C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
240: C2 = SAFMIN*MAX( ONE, BNORM )
241: C3 = BSIZE*SAFMIN
242: IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
243: C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
244: ELSE
245: C4 = ONE
246: END IF
247: IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
248: C5 = MIN( ONE, ASCALE*BSIZE )
249: ELSE
250: C5 = ONE
251: END IF
252: *
253: * Scale first eigenvalue
254: *
255: WABS = ABS( WR1 ) + ABS( WI )
256: WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
257: $ MIN( C4, HALF*MAX( WABS, C5 ) ) )
258: IF( WSIZE.NE.ONE ) THEN
259: WSCALE = ONE / WSIZE
260: IF( WSIZE.GT.ONE ) THEN
261: SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
262: $ MIN( ASCALE, BSIZE )
263: ELSE
264: SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
265: $ MAX( ASCALE, BSIZE )
266: END IF
267: WR1 = WR1*WSCALE
268: IF( WI.NE.ZERO ) THEN
269: WI = WI*WSCALE
270: WR2 = WR1
271: SCALE2 = SCALE1
272: END IF
273: ELSE
274: SCALE1 = ASCALE*BSIZE
275: SCALE2 = SCALE1
276: END IF
277: *
278: * Scale second eigenvalue (if real)
279: *
280: IF( WI.EQ.ZERO ) THEN
281: WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
282: $ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
283: IF( WSIZE.NE.ONE ) THEN
284: WSCALE = ONE / WSIZE
285: IF( WSIZE.GT.ONE ) THEN
286: SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
287: $ MIN( ASCALE, BSIZE )
288: ELSE
289: SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
290: $ MAX( ASCALE, BSIZE )
291: END IF
292: WR2 = WR2*WSCALE
293: ELSE
294: SCALE2 = ASCALE*BSIZE
295: END IF
296: END IF
297: *
298: * End of DLAG2
299: *
300: RETURN
301: END
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