1: *> \brief \b DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as necessary to avoid over-/underflow.
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DLAG2 + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlag2.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlag2.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlag2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
22: * WR2, WI )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER LDA, LDB
26: * DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), B( LDB, * )
30: * ..
31: *
32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
39: *> problem A - w B, with scaling as necessary to avoid over-/underflow.
40: *>
41: *> The scaling factor "s" results in a modified eigenvalue equation
42: *>
43: *> s A - w B
44: *>
45: *> where s is a non-negative scaling factor chosen so that w, w B,
46: *> and s A do not overflow and, if possible, do not underflow, either.
47: *> \endverbatim
48: *
49: * Arguments:
50: * ==========
51: *
52: *> \param[in] A
53: *> \verbatim
54: *> A is DOUBLE PRECISION array, dimension (LDA, 2)
55: *> On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
56: *> is less than 1/SAFMIN. Entries less than
57: *> sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
58: *> \endverbatim
59: *>
60: *> \param[in] LDA
61: *> \verbatim
62: *> LDA is INTEGER
63: *> The leading dimension of the array A. LDA >= 2.
64: *> \endverbatim
65: *>
66: *> \param[in] B
67: *> \verbatim
68: *> B is DOUBLE PRECISION array, dimension (LDB, 2)
69: *> On entry, the 2 x 2 upper triangular matrix B. It is
70: *> assumed that the one-norm of B is less than 1/SAFMIN. The
71: *> diagonals should be at least sqrt(SAFMIN) times the largest
72: *> element of B (in absolute value); if a diagonal is smaller
73: *> than that, then +/- sqrt(SAFMIN) will be used instead of
74: *> that diagonal.
75: *> \endverbatim
76: *>
77: *> \param[in] LDB
78: *> \verbatim
79: *> LDB is INTEGER
80: *> The leading dimension of the array B. LDB >= 2.
81: *> \endverbatim
82: *>
83: *> \param[in] SAFMIN
84: *> \verbatim
85: *> SAFMIN is DOUBLE PRECISION
86: *> The smallest positive number s.t. 1/SAFMIN does not
87: *> overflow. (This should always be DLAMCH('S') -- it is an
88: *> argument in order to avoid having to call DLAMCH frequently.)
89: *> \endverbatim
90: *>
91: *> \param[out] SCALE1
92: *> \verbatim
93: *> SCALE1 is DOUBLE PRECISION
94: *> A scaling factor used to avoid over-/underflow in the
95: *> eigenvalue equation which defines the first eigenvalue. If
96: *> the eigenvalues are complex, then the eigenvalues are
97: *> ( WR1 +/- WI i ) / SCALE1 (which may lie outside the
98: *> exponent range of the machine), SCALE1=SCALE2, and SCALE1
99: *> will always be positive. If the eigenvalues are real, then
100: *> the first (real) eigenvalue is WR1 / SCALE1 , but this may
101: *> overflow or underflow, and in fact, SCALE1 may be zero or
102: *> less than the underflow threshold if the exact eigenvalue
103: *> is sufficiently large.
104: *> \endverbatim
105: *>
106: *> \param[out] SCALE2
107: *> \verbatim
108: *> SCALE2 is DOUBLE PRECISION
109: *> A scaling factor used to avoid over-/underflow in the
110: *> eigenvalue equation which defines the second eigenvalue. If
111: *> the eigenvalues are complex, then SCALE2=SCALE1. If the
112: *> eigenvalues are real, then the second (real) eigenvalue is
113: *> WR2 / SCALE2 , but this may overflow or underflow, and in
114: *> fact, SCALE2 may be zero or less than the underflow
115: *> threshold if the exact eigenvalue is sufficiently large.
116: *> \endverbatim
117: *>
118: *> \param[out] WR1
119: *> \verbatim
120: *> WR1 is DOUBLE PRECISION
121: *> If the eigenvalue is real, then WR1 is SCALE1 times the
122: *> eigenvalue closest to the (2,2) element of A B**(-1). If the
123: *> eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
124: *> part of the eigenvalues.
125: *> \endverbatim
126: *>
127: *> \param[out] WR2
128: *> \verbatim
129: *> WR2 is DOUBLE PRECISION
130: *> If the eigenvalue is real, then WR2 is SCALE2 times the
131: *> other eigenvalue. If the eigenvalue is complex, then
132: *> WR1=WR2 is SCALE1 times the real part of the eigenvalues.
133: *> \endverbatim
134: *>
135: *> \param[out] WI
136: *> \verbatim
137: *> WI is DOUBLE PRECISION
138: *> If the eigenvalue is real, then WI is zero. If the
139: *> eigenvalue is complex, then WI is SCALE1 times the imaginary
140: *> part of the eigenvalues. WI will always be non-negative.
141: *> \endverbatim
142: *
143: * Authors:
144: * ========
145: *
146: *> \author Univ. of Tennessee
147: *> \author Univ. of California Berkeley
148: *> \author Univ. of Colorado Denver
149: *> \author NAG Ltd.
150: *
151: *> \ingroup doubleOTHERauxiliary
152: *
153: * =====================================================================
154: SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
155: $ WR2, WI )
156: *
157: * -- LAPACK auxiliary routine --
158: * -- LAPACK is a software package provided by Univ. of Tennessee, --
159: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
160: *
161: * .. Scalar Arguments ..
162: INTEGER LDA, LDB
163: DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
164: * ..
165: * .. Array Arguments ..
166: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
167: * ..
168: *
169: * =====================================================================
170: *
171: * .. Parameters ..
172: DOUBLE PRECISION ZERO, ONE, TWO
173: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
174: DOUBLE PRECISION HALF
175: PARAMETER ( HALF = ONE / TWO )
176: DOUBLE PRECISION FUZZY1
177: PARAMETER ( FUZZY1 = ONE+1.0D-5 )
178: * ..
179: * .. Local Scalars ..
180: DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
181: $ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
182: $ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
183: $ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
184: $ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
185: $ WSCALE, WSIZE, WSMALL
186: * ..
187: * .. Intrinsic Functions ..
188: INTRINSIC ABS, MAX, MIN, SIGN, SQRT
189: * ..
190: * .. Executable Statements ..
191: *
192: RTMIN = SQRT( SAFMIN )
193: RTMAX = ONE / RTMIN
194: SAFMAX = ONE / SAFMIN
195: *
196: * Scale A
197: *
198: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
199: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
200: ASCALE = ONE / ANORM
201: A11 = ASCALE*A( 1, 1 )
202: A21 = ASCALE*A( 2, 1 )
203: A12 = ASCALE*A( 1, 2 )
204: A22 = ASCALE*A( 2, 2 )
205: *
206: * Perturb B if necessary to insure non-singularity
207: *
208: B11 = B( 1, 1 )
209: B12 = B( 1, 2 )
210: B22 = B( 2, 2 )
211: BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
212: IF( ABS( B11 ).LT.BMIN )
213: $ B11 = SIGN( BMIN, B11 )
214: IF( ABS( B22 ).LT.BMIN )
215: $ B22 = SIGN( BMIN, B22 )
216: *
217: * Scale B
218: *
219: BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
220: BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
221: BSCALE = ONE / BSIZE
222: B11 = B11*BSCALE
223: B12 = B12*BSCALE
224: B22 = B22*BSCALE
225: *
226: * Compute larger eigenvalue by method described by C. van Loan
227: *
228: * ( AS is A shifted by -SHIFT*B )
229: *
230: BINV11 = ONE / B11
231: BINV22 = ONE / B22
232: S1 = A11*BINV11
233: S2 = A22*BINV22
234: IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
235: AS12 = A12 - S1*B12
236: AS22 = A22 - S1*B22
237: SS = A21*( BINV11*BINV22 )
238: ABI22 = AS22*BINV22 - SS*B12
239: PP = HALF*ABI22
240: SHIFT = S1
241: ELSE
242: AS12 = A12 - S2*B12
243: AS11 = A11 - S2*B11
244: SS = A21*( BINV11*BINV22 )
245: ABI22 = -SS*B12
246: PP = HALF*( AS11*BINV11+ABI22 )
247: SHIFT = S2
248: END IF
249: QQ = SS*AS12
250: IF( ABS( PP*RTMIN ).GE.ONE ) THEN
251: DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
252: R = SQRT( ABS( DISCR ) )*RTMAX
253: ELSE
254: IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
255: DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
256: R = SQRT( ABS( DISCR ) )*RTMIN
257: ELSE
258: DISCR = PP**2 + QQ
259: R = SQRT( ABS( DISCR ) )
260: END IF
261: END IF
262: *
263: * Note: the test of R in the following IF is to cover the case when
264: * DISCR is small and negative and is flushed to zero during
265: * the calculation of R. On machines which have a consistent
266: * flush-to-zero threshold and handle numbers above that
267: * threshold correctly, it would not be necessary.
268: *
269: IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
270: SUM = PP + SIGN( R, PP )
271: DIFF = PP - SIGN( R, PP )
272: WBIG = SHIFT + SUM
273: *
274: * Compute smaller eigenvalue
275: *
276: WSMALL = SHIFT + DIFF
277: IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
278: WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
279: WSMALL = WDET / WBIG
280: END IF
281: *
282: * Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
283: * for WR1.
284: *
285: IF( PP.GT.ABI22 ) THEN
286: WR1 = MIN( WBIG, WSMALL )
287: WR2 = MAX( WBIG, WSMALL )
288: ELSE
289: WR1 = MAX( WBIG, WSMALL )
290: WR2 = MIN( WBIG, WSMALL )
291: END IF
292: WI = ZERO
293: ELSE
294: *
295: * Complex eigenvalues
296: *
297: WR1 = SHIFT + PP
298: WR2 = WR1
299: WI = R
300: END IF
301: *
302: * Further scaling to avoid underflow and overflow in computing
303: * SCALE1 and overflow in computing w*B.
304: *
305: * This scale factor (WSCALE) is bounded from above using C1 and C2,
306: * and from below using C3 and C4.
307: * C1 implements the condition s A must never overflow.
308: * C2 implements the condition w B must never overflow.
309: * C3, with C2,
310: * implement the condition that s A - w B must never overflow.
311: * C4 implements the condition s should not underflow.
312: * C5 implements the condition max(s,|w|) should be at least 2.
313: *
314: C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
315: C2 = SAFMIN*MAX( ONE, BNORM )
316: C3 = BSIZE*SAFMIN
317: IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
318: C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
319: ELSE
320: C4 = ONE
321: END IF
322: IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
323: C5 = MIN( ONE, ASCALE*BSIZE )
324: ELSE
325: C5 = ONE
326: END IF
327: *
328: * Scale first eigenvalue
329: *
330: WABS = ABS( WR1 ) + ABS( WI )
331: WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
332: $ MIN( C4, HALF*MAX( WABS, C5 ) ) )
333: IF( WSIZE.NE.ONE ) THEN
334: WSCALE = ONE / WSIZE
335: IF( WSIZE.GT.ONE ) THEN
336: SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
337: $ MIN( ASCALE, BSIZE )
338: ELSE
339: SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
340: $ MAX( ASCALE, BSIZE )
341: END IF
342: WR1 = WR1*WSCALE
343: IF( WI.NE.ZERO ) THEN
344: WI = WI*WSCALE
345: WR2 = WR1
346: SCALE2 = SCALE1
347: END IF
348: ELSE
349: SCALE1 = ASCALE*BSIZE
350: SCALE2 = SCALE1
351: END IF
352: *
353: * Scale second eigenvalue (if real)
354: *
355: IF( WI.EQ.ZERO ) THEN
356: WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
357: $ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
358: IF( WSIZE.NE.ONE ) THEN
359: WSCALE = ONE / WSIZE
360: IF( WSIZE.GT.ONE ) THEN
361: SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
362: $ MIN( ASCALE, BSIZE )
363: ELSE
364: SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
365: $ MAX( ASCALE, BSIZE )
366: END IF
367: WR2 = WR2*WSCALE
368: ELSE
369: SCALE2 = ASCALE*BSIZE
370: END IF
371: END IF
372: *
373: * End of DLAG2
374: *
375: RETURN
376: END
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