LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cheequb.f
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1 *> \brief \b CHEEQUB
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE CHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, N
25 * REAL AMAX, SCOND
26 * CHARACTER UPLO
27 * ..
28 * .. Array Arguments ..
29 * COMPLEX A( LDA, * ), WORK( * )
30 * REAL S( * )
31 * ..
32 *
33 *
34 *> \par Purpose:
35 * =============
36 *>
37 *> \verbatim
38 *>
39 *> CHEEQUB computes row and column scalings intended to equilibrate a
40 *> Hermitian matrix A (with respect to the Euclidean norm) and reduce
41 *> its condition number. The scale factors S are computed by the BIN
42 *> algorithm (see references) so that the scaled matrix B with elements
43 *> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
44 *> the smallest possible condition number over all possible diagonal
45 *> scalings.
46 *> \endverbatim
47 *
48 * Arguments:
49 * ==========
50 *
51 *> \param[in] UPLO
52 *> \verbatim
53 *> UPLO is CHARACTER*1
54 *> = 'U': Upper triangle of A is stored;
55 *> = 'L': Lower triangle of A is stored.
56 *> \endverbatim
57 *>
58 *> \param[in] N
59 *> \verbatim
60 *> N is INTEGER
61 *> The order of the matrix A. N >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in] A
65 *> \verbatim
66 *> A is COMPLEX array, dimension (LDA,N)
67 *> The N-by-N Hermitian matrix whose scaling factors are to be
68 *> computed.
69 *> \endverbatim
70 *>
71 *> \param[in] LDA
72 *> \verbatim
73 *> LDA is INTEGER
74 *> The leading dimension of the array A. LDA >= max(1,N).
75 *> \endverbatim
76 *>
77 *> \param[out] S
78 *> \verbatim
79 *> S is REAL array, dimension (N)
80 *> If INFO = 0, S contains the scale factors for A.
81 *> \endverbatim
82 *>
83 *> \param[out] SCOND
84 *> \verbatim
85 *> SCOND is REAL
86 *> If INFO = 0, S contains the ratio of the smallest S(i) to
87 *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
88 *> large nor too small, it is not worth scaling by S.
89 *> \endverbatim
90 *>
91 *> \param[out] AMAX
92 *> \verbatim
93 *> AMAX is REAL
94 *> Largest absolute value of any matrix element. If AMAX is
95 *> very close to overflow or very close to underflow, the
96 *> matrix should be scaled.
97 *> \endverbatim
98 *>
99 *> \param[out] WORK
100 *> \verbatim
101 *> WORK is COMPLEX array, dimension (2*N)
102 *> \endverbatim
103 *>
104 *> \param[out] INFO
105 *> \verbatim
106 *> INFO is INTEGER
107 *> = 0: successful exit
108 *> < 0: if INFO = -i, the i-th argument had an illegal value
109 *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
110 *> \endverbatim
111 *
112 * Authors:
113 * ========
114 *
115 *> \author Univ. of Tennessee
116 *> \author Univ. of California Berkeley
117 *> \author Univ. of Colorado Denver
118 *> \author NAG Ltd.
119 *
120 *> \ingroup complexHEcomputational
121 *
122 *> \par References:
123 * ================
124 *>
125 *> Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
126 *> Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
127 *> DOI 10.1023/B:NUMA.0000016606.32820.69 \n
128 *> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
129 *>
130 * =====================================================================
131  SUBROUTINE cheequb( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
132 *
133 * -- LAPACK computational routine --
134 * -- LAPACK is a software package provided by Univ. of Tennessee, --
135 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136 *
137 * .. Scalar Arguments ..
138  INTEGER INFO, LDA, N
139  REAL AMAX, SCOND
140  CHARACTER UPLO
141 * ..
142 * .. Array Arguments ..
143  COMPLEX A( LDA, * ), WORK( * )
144  REAL S( * )
145 * ..
146 *
147 * =====================================================================
148 *
149 * .. Parameters ..
150  REAL ONE, ZERO
151  parameter( one = 1.0e0, zero = 0.0e0 )
152  INTEGER MAX_ITER
153  parameter( max_iter = 100 )
154 * ..
155 * .. Local Scalars ..
156  INTEGER I, J, ITER
157  REAL AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
158  $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
159  LOGICAL UP
160  COMPLEX ZDUM
161 * ..
162 * .. External Functions ..
163  REAL SLAMCH
164  LOGICAL LSAME
165  EXTERNAL lsame, slamch
166 * ..
167 * .. External Subroutines ..
168  EXTERNAL classq, xerbla
169 * ..
170 * .. Intrinsic Functions ..
171  INTRINSIC abs, aimag, int, log, max, min, real, sqrt
172 * ..
173 * .. Statement Functions ..
174  REAL CABS1
175 * ..
176 * .. Statement Function Definitions ..
177  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
178 * ..
179 * .. Executable Statements ..
180 *
181 * Test the input parameters.
182 *
183  info = 0
184  IF ( .NOT. ( lsame( uplo, 'U' ) .OR. lsame( uplo, 'L' ) ) ) THEN
185  info = -1
186  ELSE IF ( n .LT. 0 ) THEN
187  info = -2
188  ELSE IF ( lda .LT. max( 1, n ) ) THEN
189  info = -4
190  END IF
191  IF ( info .NE. 0 ) THEN
192  CALL xerbla( 'CHEEQUB', -info )
193  RETURN
194  END IF
195 
196  up = lsame( uplo, 'U' )
197  amax = zero
198 *
199 * Quick return if possible.
200 *
201  IF ( n .EQ. 0 ) THEN
202  scond = one
203  RETURN
204  END IF
205 
206  DO i = 1, n
207  s( i ) = zero
208  END DO
209 
210  amax = zero
211  IF ( up ) THEN
212  DO j = 1, n
213  DO i = 1, j-1
214  s( i ) = max( s( i ), cabs1( a( i, j ) ) )
215  s( j ) = max( s( j ), cabs1( a( i, j ) ) )
216  amax = max( amax, cabs1( a( i, j ) ) )
217  END DO
218  s( j ) = max( s( j ), cabs1( a( j, j ) ) )
219  amax = max( amax, cabs1( a( j, j ) ) )
220  END DO
221  ELSE
222  DO j = 1, n
223  s( j ) = max( s( j ), cabs1( a( j, j ) ) )
224  amax = max( amax, cabs1( a( j, j ) ) )
225  DO i = j+1, n
226  s( i ) = max( s( i ), cabs1( a( i, j ) ) )
227  s( j ) = max( s( j ), cabs1( a( i, j ) ) )
228  amax = max( amax, cabs1( a( i, j ) ) )
229  END DO
230  END DO
231  END IF
232  DO j = 1, n
233  s( j ) = 1.0e0 / s( j )
234  END DO
235 
236  tol = one / sqrt( 2.0e0 * n )
237 
238  DO iter = 1, max_iter
239  scale = 0.0e0
240  sumsq = 0.0e0
241 * beta = |A|s
242  DO i = 1, n
243  work( i ) = zero
244  END DO
245  IF ( up ) THEN
246  DO j = 1, n
247  DO i = 1, j-1
248  work( i ) = work( i ) + cabs1( a( i, j ) ) * s( j )
249  work( j ) = work( j ) + cabs1( a( i, j ) ) * s( i )
250  END DO
251  work( j ) = work( j ) + cabs1( a( j, j ) ) * s( j )
252  END DO
253  ELSE
254  DO j = 1, n
255  work( j ) = work( j ) + cabs1( a( j, j ) ) * s( j )
256  DO i = j+1, n
257  work( i ) = work( i ) + cabs1( a( i, j ) ) * s( j )
258  work( j ) = work( j ) + cabs1( a( i, j ) ) * s( i )
259  END DO
260  END DO
261  END IF
262 
263 * avg = s^T beta / n
264  avg = 0.0e0
265  DO i = 1, n
266  avg = avg + real( s( i )*work( i ) )
267  END DO
268  avg = avg / n
269 
270  std = 0.0e0
271  DO i = n+1, 2*n
272  work( i ) = s( i-n ) * work( i-n ) - avg
273  END DO
274  CALL classq( n, work( n+1 ), 1, scale, sumsq )
275  std = scale * sqrt( sumsq / n )
276 
277  IF ( std .LT. tol * avg ) GOTO 999
278 
279  DO i = 1, n
280  t = cabs1( a( i, i ) )
281  si = s( i )
282  c2 = ( n-1 ) * t
283  c1 = real( ( n-2 ) * ( work( i ) - t*si ) )
284  c0 = real( -(t*si)*si + 2*work( i )*si - n*avg )
285  d = c1*c1 - 4*c0*c2
286 
287  IF ( d .LE. 0 ) THEN
288  info = -1
289  RETURN
290  END IF
291  si = -2*c0 / ( c1 + sqrt( d ) )
292 
293  d = si - s( i )
294  u = zero
295  IF ( up ) THEN
296  DO j = 1, i
297  t = cabs1( a( j, i ) )
298  u = u + s( j )*t
299  work( j ) = work( j ) + d*t
300  END DO
301  DO j = i+1,n
302  t = cabs1( a( i, j ) )
303  u = u + s( j )*t
304  work( j ) = work( j ) + d*t
305  END DO
306  ELSE
307  DO j = 1, i
308  t = cabs1( a( i, j ) )
309  u = u + s( j )*t
310  work( j ) = work( j ) + d*t
311  END DO
312  DO j = i+1,n
313  t = cabs1( a( j, i ) )
314  u = u + s( j )*t
315  work( j ) = work( j ) + d*t
316  END DO
317  END IF
318 
319  avg = avg + real( ( u + work( i ) ) * d / n )
320  s( i ) = si
321  END DO
322  END DO
323 
324  999 CONTINUE
325 
326  smlnum = slamch( 'SAFEMIN' )
327  bignum = one / smlnum
328  smin = bignum
329  smax = zero
330  t = one / sqrt( avg )
331  base = slamch( 'B' )
332  u = one / log( base )
333  DO i = 1, n
334  s( i ) = base ** int( u * log( s( i ) * t ) )
335  smin = min( smin, s( i ) )
336  smax = max( smax, s( i ) )
337  END DO
338  scond = max( smin, smlnum ) / min( smax, bignum )
339 *
340  END
subroutine classq(n, x, incx, scl, sumsq)
CLASSQ updates a sum of squares represented in scaled form.
Definition: classq.f90:126
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine cheequb(UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO)
CHEEQUB
Definition: cheequb.f:132