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: *> \date June 2016
152: *
153: *> \ingroup doubleOTHERauxiliary
154: *
155: * =====================================================================
156: SUBROUTINE DLAG2( A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1,
157: $ WR2, WI )
158: *
159: * -- LAPACK auxiliary routine (version 3.7.0) --
160: * -- LAPACK is a software package provided by Univ. of Tennessee, --
161: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
162: * June 2016
163: *
164: * .. Scalar Arguments ..
165: INTEGER LDA, LDB
166: DOUBLE PRECISION SAFMIN, SCALE1, SCALE2, WI, WR1, WR2
167: * ..
168: * .. Array Arguments ..
169: DOUBLE PRECISION A( LDA, * ), B( LDB, * )
170: * ..
171: *
172: * =====================================================================
173: *
174: * .. Parameters ..
175: DOUBLE PRECISION ZERO, ONE, TWO
176: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
177: DOUBLE PRECISION HALF
178: PARAMETER ( HALF = ONE / TWO )
179: DOUBLE PRECISION FUZZY1
180: PARAMETER ( FUZZY1 = ONE+1.0D-5 )
181: * ..
182: * .. Local Scalars ..
183: DOUBLE PRECISION A11, A12, A21, A22, ABI22, ANORM, AS11, AS12,
184: $ AS22, ASCALE, B11, B12, B22, BINV11, BINV22,
185: $ BMIN, BNORM, BSCALE, BSIZE, C1, C2, C3, C4, C5,
186: $ DIFF, DISCR, PP, QQ, R, RTMAX, RTMIN, S1, S2,
187: $ SAFMAX, SHIFT, SS, SUM, WABS, WBIG, WDET,
188: $ WSCALE, WSIZE, WSMALL
189: * ..
190: * .. Intrinsic Functions ..
191: INTRINSIC ABS, MAX, MIN, SIGN, SQRT
192: * ..
193: * .. Executable Statements ..
194: *
195: RTMIN = SQRT( SAFMIN )
196: RTMAX = ONE / RTMIN
197: SAFMAX = ONE / SAFMIN
198: *
199: * Scale A
200: *
201: ANORM = MAX( ABS( A( 1, 1 ) )+ABS( A( 2, 1 ) ),
202: $ ABS( A( 1, 2 ) )+ABS( A( 2, 2 ) ), SAFMIN )
203: ASCALE = ONE / ANORM
204: A11 = ASCALE*A( 1, 1 )
205: A21 = ASCALE*A( 2, 1 )
206: A12 = ASCALE*A( 1, 2 )
207: A22 = ASCALE*A( 2, 2 )
208: *
209: * Perturb B if necessary to insure non-singularity
210: *
211: B11 = B( 1, 1 )
212: B12 = B( 1, 2 )
213: B22 = B( 2, 2 )
214: BMIN = RTMIN*MAX( ABS( B11 ), ABS( B12 ), ABS( B22 ), RTMIN )
215: IF( ABS( B11 ).LT.BMIN )
216: $ B11 = SIGN( BMIN, B11 )
217: IF( ABS( B22 ).LT.BMIN )
218: $ B22 = SIGN( BMIN, B22 )
219: *
220: * Scale B
221: *
222: BNORM = MAX( ABS( B11 ), ABS( B12 )+ABS( B22 ), SAFMIN )
223: BSIZE = MAX( ABS( B11 ), ABS( B22 ) )
224: BSCALE = ONE / BSIZE
225: B11 = B11*BSCALE
226: B12 = B12*BSCALE
227: B22 = B22*BSCALE
228: *
229: * Compute larger eigenvalue by method described by C. van Loan
230: *
231: * ( AS is A shifted by -SHIFT*B )
232: *
233: BINV11 = ONE / B11
234: BINV22 = ONE / B22
235: S1 = A11*BINV11
236: S2 = A22*BINV22
237: IF( ABS( S1 ).LE.ABS( S2 ) ) THEN
238: AS12 = A12 - S1*B12
239: AS22 = A22 - S1*B22
240: SS = A21*( BINV11*BINV22 )
241: ABI22 = AS22*BINV22 - SS*B12
242: PP = HALF*ABI22
243: SHIFT = S1
244: ELSE
245: AS12 = A12 - S2*B12
246: AS11 = A11 - S2*B11
247: SS = A21*( BINV11*BINV22 )
248: ABI22 = -SS*B12
249: PP = HALF*( AS11*BINV11+ABI22 )
250: SHIFT = S2
251: END IF
252: QQ = SS*AS12
253: IF( ABS( PP*RTMIN ).GE.ONE ) THEN
254: DISCR = ( RTMIN*PP )**2 + QQ*SAFMIN
255: R = SQRT( ABS( DISCR ) )*RTMAX
256: ELSE
257: IF( PP**2+ABS( QQ ).LE.SAFMIN ) THEN
258: DISCR = ( RTMAX*PP )**2 + QQ*SAFMAX
259: R = SQRT( ABS( DISCR ) )*RTMIN
260: ELSE
261: DISCR = PP**2 + QQ
262: R = SQRT( ABS( DISCR ) )
263: END IF
264: END IF
265: *
266: * Note: the test of R in the following IF is to cover the case when
267: * DISCR is small and negative and is flushed to zero during
268: * the calculation of R. On machines which have a consistent
269: * flush-to-zero threshold and handle numbers above that
270: * threshold correctly, it would not be necessary.
271: *
272: IF( DISCR.GE.ZERO .OR. R.EQ.ZERO ) THEN
273: SUM = PP + SIGN( R, PP )
274: DIFF = PP - SIGN( R, PP )
275: WBIG = SHIFT + SUM
276: *
277: * Compute smaller eigenvalue
278: *
279: WSMALL = SHIFT + DIFF
280: IF( HALF*ABS( WBIG ).GT.MAX( ABS( WSMALL ), SAFMIN ) ) THEN
281: WDET = ( A11*A22-A12*A21 )*( BINV11*BINV22 )
282: WSMALL = WDET / WBIG
283: END IF
284: *
285: * Choose (real) eigenvalue closest to 2,2 element of A*B**(-1)
286: * for WR1.
287: *
288: IF( PP.GT.ABI22 ) THEN
289: WR1 = MIN( WBIG, WSMALL )
290: WR2 = MAX( WBIG, WSMALL )
291: ELSE
292: WR1 = MAX( WBIG, WSMALL )
293: WR2 = MIN( WBIG, WSMALL )
294: END IF
295: WI = ZERO
296: ELSE
297: *
298: * Complex eigenvalues
299: *
300: WR1 = SHIFT + PP
301: WR2 = WR1
302: WI = R
303: END IF
304: *
305: * Further scaling to avoid underflow and overflow in computing
306: * SCALE1 and overflow in computing w*B.
307: *
308: * This scale factor (WSCALE) is bounded from above using C1 and C2,
309: * and from below using C3 and C4.
310: * C1 implements the condition s A must never overflow.
311: * C2 implements the condition w B must never overflow.
312: * C3, with C2,
313: * implement the condition that s A - w B must never overflow.
314: * C4 implements the condition s should not underflow.
315: * C5 implements the condition max(s,|w|) should be at least 2.
316: *
317: C1 = BSIZE*( SAFMIN*MAX( ONE, ASCALE ) )
318: C2 = SAFMIN*MAX( ONE, BNORM )
319: C3 = BSIZE*SAFMIN
320: IF( ASCALE.LE.ONE .AND. BSIZE.LE.ONE ) THEN
321: C4 = MIN( ONE, ( ASCALE / SAFMIN )*BSIZE )
322: ELSE
323: C4 = ONE
324: END IF
325: IF( ASCALE.LE.ONE .OR. BSIZE.LE.ONE ) THEN
326: C5 = MIN( ONE, ASCALE*BSIZE )
327: ELSE
328: C5 = ONE
329: END IF
330: *
331: * Scale first eigenvalue
332: *
333: WABS = ABS( WR1 ) + ABS( WI )
334: WSIZE = MAX( SAFMIN, C1, FUZZY1*( WABS*C2+C3 ),
335: $ MIN( C4, HALF*MAX( WABS, C5 ) ) )
336: IF( WSIZE.NE.ONE ) THEN
337: WSCALE = ONE / WSIZE
338: IF( WSIZE.GT.ONE ) THEN
339: SCALE1 = ( MAX( ASCALE, BSIZE )*WSCALE )*
340: $ MIN( ASCALE, BSIZE )
341: ELSE
342: SCALE1 = ( MIN( ASCALE, BSIZE )*WSCALE )*
343: $ MAX( ASCALE, BSIZE )
344: END IF
345: WR1 = WR1*WSCALE
346: IF( WI.NE.ZERO ) THEN
347: WI = WI*WSCALE
348: WR2 = WR1
349: SCALE2 = SCALE1
350: END IF
351: ELSE
352: SCALE1 = ASCALE*BSIZE
353: SCALE2 = SCALE1
354: END IF
355: *
356: * Scale second eigenvalue (if real)
357: *
358: IF( WI.EQ.ZERO ) THEN
359: WSIZE = MAX( SAFMIN, C1, FUZZY1*( ABS( WR2 )*C2+C3 ),
360: $ MIN( C4, HALF*MAX( ABS( WR2 ), C5 ) ) )
361: IF( WSIZE.NE.ONE ) THEN
362: WSCALE = ONE / WSIZE
363: IF( WSIZE.GT.ONE ) THEN
364: SCALE2 = ( MAX( ASCALE, BSIZE )*WSCALE )*
365: $ MIN( ASCALE, BSIZE )
366: ELSE
367: SCALE2 = ( MIN( ASCALE, BSIZE )*WSCALE )*
368: $ MAX( ASCALE, BSIZE )
369: END IF
370: WR2 = WR2*WSCALE
371: ELSE
372: SCALE2 = ASCALE*BSIZE
373: END IF
374: END IF
375: *
376: * End of DLAG2
377: *
378: RETURN
379: END
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