Annotation of rpl/lapack/lapack/dsyequb.f, revision 1.11
1.5 bertrand 1: *> \brief \b DSYEQUB
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.11 ! bertrand 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
1.5 bertrand 7: *
8: *> \htmlonly
1.11 ! bertrand 9: *> Download DSYEQUB + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsyequb.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsyequb.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsyequb.f">
1.5 bertrand 15: *> [TXT]</a>
1.11 ! bertrand 16: *> \endhtmlonly
1.5 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
1.11 ! bertrand 22: *
1.5 bertrand 23: * .. Scalar Arguments ..
24: * INTEGER INFO, LDA, N
25: * DOUBLE PRECISION AMAX, SCOND
26: * CHARACTER UPLO
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
30: * ..
1.11 ! bertrand 31: *
1.5 bertrand 32: *
33: *> \par Purpose:
34: * =============
35: *>
36: *> \verbatim
37: *>
38: *> DSYEQUB computes row and column scalings intended to equilibrate a
1.11 ! bertrand 39: *> symmetric matrix A (with respect to the Euclidean norm) and reduce
! 40: *> its condition number. The scale factors S are computed by the BIN
! 41: *> algorithm (see references) so that the scaled matrix B with elements
! 42: *> B(i,j) = S(i)*A(i,j)*S(j) has a condition number within a factor N of
! 43: *> the smallest possible condition number over all possible diagonal
1.5 bertrand 44: *> scalings.
45: *> \endverbatim
46: *
47: * Arguments:
48: * ==========
49: *
50: *> \param[in] UPLO
51: *> \verbatim
52: *> UPLO is CHARACTER*1
1.11 ! bertrand 53: *> = 'U': Upper triangle of A is stored;
! 54: *> = 'L': Lower triangle of A is stored.
1.5 bertrand 55: *> \endverbatim
56: *>
57: *> \param[in] N
58: *> \verbatim
59: *> N is INTEGER
1.11 ! bertrand 60: *> The order of the matrix A. N >= 0.
1.5 bertrand 61: *> \endverbatim
62: *>
63: *> \param[in] A
64: *> \verbatim
65: *> A is DOUBLE PRECISION array, dimension (LDA,N)
1.11 ! bertrand 66: *> The N-by-N symmetric matrix whose scaling factors are to be
! 67: *> computed.
1.5 bertrand 68: *> \endverbatim
69: *>
70: *> \param[in] LDA
71: *> \verbatim
72: *> LDA is INTEGER
1.11 ! bertrand 73: *> The leading dimension of the array A. LDA >= max(1,N).
1.5 bertrand 74: *> \endverbatim
75: *>
76: *> \param[out] S
77: *> \verbatim
78: *> S is DOUBLE PRECISION array, dimension (N)
79: *> If INFO = 0, S contains the scale factors for A.
80: *> \endverbatim
81: *>
82: *> \param[out] SCOND
83: *> \verbatim
84: *> SCOND is DOUBLE PRECISION
85: *> If INFO = 0, S contains the ratio of the smallest S(i) to
1.11 ! bertrand 86: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
1.5 bertrand 87: *> large nor too small, it is not worth scaling by S.
88: *> \endverbatim
89: *>
90: *> \param[out] AMAX
91: *> \verbatim
92: *> AMAX is DOUBLE PRECISION
1.11 ! bertrand 93: *> Largest absolute value of any matrix element. If AMAX is
! 94: *> very close to overflow or very close to underflow, the
! 95: *> matrix should be scaled.
1.5 bertrand 96: *> \endverbatim
97: *>
98: *> \param[out] WORK
99: *> \verbatim
1.11 ! bertrand 100: *> WORK is DOUBLE PRECISION array, dimension (2*N)
1.5 bertrand 101: *> \endverbatim
102: *>
103: *> \param[out] INFO
104: *> \verbatim
105: *> INFO is INTEGER
106: *> = 0: successful exit
107: *> < 0: if INFO = -i, the i-th argument had an illegal value
108: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
109: *> \endverbatim
110: *
111: * Authors:
112: * ========
113: *
1.11 ! bertrand 114: *> \author Univ. of Tennessee
! 115: *> \author Univ. of California Berkeley
! 116: *> \author Univ. of Colorado Denver
! 117: *> \author NAG Ltd.
1.5 bertrand 118: *
1.11 ! bertrand 119: *> \date December 2016
1.5 bertrand 120: *
121: *> \ingroup doubleSYcomputational
122: *
123: *> \par References:
124: * ================
125: *>
126: *> Livne, O.E. and Golub, G.H., "Scaling by Binormalization", \n
127: *> Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. \n
128: *> DOI 10.1023/B:NUMA.0000016606.32820.69 \n
1.11 ! bertrand 129: *> Tech report version: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.3.1679
1.5 bertrand 130: *>
131: * =====================================================================
1.1 bertrand 132: SUBROUTINE DSYEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFO )
133: *
1.11 ! bertrand 134: * -- LAPACK computational routine (version 3.7.0) --
1.5 bertrand 135: * -- LAPACK is a software package provided by Univ. of Tennessee, --
136: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.11 ! bertrand 137: * December 2016
1.1 bertrand 138: *
139: * .. Scalar Arguments ..
140: INTEGER INFO, LDA, N
141: DOUBLE PRECISION AMAX, SCOND
142: CHARACTER UPLO
143: * ..
144: * .. Array Arguments ..
145: DOUBLE PRECISION A( LDA, * ), S( * ), WORK( * )
146: * ..
147: *
148: * =====================================================================
149: *
150: * .. Parameters ..
151: DOUBLE PRECISION ONE, ZERO
1.11 ! bertrand 152: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0 )
1.1 bertrand 153: INTEGER MAX_ITER
154: PARAMETER ( MAX_ITER = 100 )
155: * ..
156: * .. Local Scalars ..
157: INTEGER I, J, ITER
158: DOUBLE PRECISION AVG, STD, TOL, C0, C1, C2, T, U, SI, D, BASE,
159: $ SMIN, SMAX, SMLNUM, BIGNUM, SCALE, SUMSQ
160: LOGICAL UP
161: * ..
162: * .. External Functions ..
163: DOUBLE PRECISION DLAMCH
164: LOGICAL LSAME
165: EXTERNAL DLAMCH, LSAME
166: * ..
167: * .. External Subroutines ..
168: EXTERNAL DLASSQ
169: * ..
170: * .. Intrinsic Functions ..
171: INTRINSIC ABS, INT, LOG, MAX, MIN, SQRT
172: * ..
173: * .. Executable Statements ..
174: *
1.11 ! bertrand 175: * Test the input parameters.
1.1 bertrand 176: *
177: INFO = 0
178: IF ( .NOT. ( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) ) THEN
1.11 ! bertrand 179: INFO = -1
1.1 bertrand 180: ELSE IF ( N .LT. 0 ) THEN
1.11 ! bertrand 181: INFO = -2
1.1 bertrand 182: ELSE IF ( LDA .LT. MAX( 1, N ) ) THEN
1.11 ! bertrand 183: INFO = -4
1.1 bertrand 184: END IF
185: IF ( INFO .NE. 0 ) THEN
1.11 ! bertrand 186: CALL XERBLA( 'DSYEQUB', -INFO )
! 187: RETURN
1.1 bertrand 188: END IF
189:
190: UP = LSAME( UPLO, 'U' )
191: AMAX = ZERO
192: *
193: * Quick return if possible.
194: *
195: IF ( N .EQ. 0 ) THEN
1.11 ! bertrand 196: SCOND = ONE
! 197: RETURN
1.1 bertrand 198: END IF
199:
200: DO I = 1, N
1.11 ! bertrand 201: S( I ) = ZERO
1.1 bertrand 202: END DO
203:
204: AMAX = ZERO
205: IF ( UP ) THEN
206: DO J = 1, N
207: DO I = 1, J-1
208: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
209: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
1.11 ! bertrand 210: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
1.1 bertrand 211: END DO
212: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
213: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
214: END DO
215: ELSE
216: DO J = 1, N
217: S( J ) = MAX( S( J ), ABS( A( J, J ) ) )
218: AMAX = MAX( AMAX, ABS( A( J, J ) ) )
219: DO I = J+1, N
220: S( I ) = MAX( S( I ), ABS( A( I, J ) ) )
221: S( J ) = MAX( S( J ), ABS( A( I, J ) ) )
222: AMAX = MAX( AMAX, ABS( A( I, J ) ) )
223: END DO
224: END DO
225: END IF
226: DO J = 1, N
1.11 ! bertrand 227: S( J ) = 1.0D0 / S( J )
1.1 bertrand 228: END DO
229:
1.11 ! bertrand 230: TOL = ONE / SQRT( 2.0D0 * N )
1.1 bertrand 231:
232: DO ITER = 1, MAX_ITER
1.11 ! bertrand 233: SCALE = 0.0D0
! 234: SUMSQ = 0.0D0
! 235: * beta = |A|s
! 236: DO I = 1, N
! 237: WORK( I ) = ZERO
! 238: END DO
! 239: IF ( UP ) THEN
! 240: DO J = 1, N
! 241: DO I = 1, J-1
! 242: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
! 243: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
! 244: END DO
! 245: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
1.1 bertrand 246: END DO
1.11 ! bertrand 247: ELSE
! 248: DO J = 1, N
! 249: WORK( J ) = WORK( J ) + ABS( A( J, J ) ) * S( J )
! 250: DO I = J+1, N
! 251: WORK( I ) = WORK( I ) + ABS( A( I, J ) ) * S( J )
! 252: WORK( J ) = WORK( J ) + ABS( A( I, J ) ) * S( I )
! 253: END DO
1.1 bertrand 254: END DO
1.11 ! bertrand 255: END IF
! 256:
! 257: * avg = s^T beta / n
! 258: AVG = 0.0D0
! 259: DO I = 1, N
! 260: AVG = AVG + S( I )*WORK( I )
! 261: END DO
! 262: AVG = AVG / N
! 263:
! 264: STD = 0.0D0
! 265: DO I = N+1, 2*N
! 266: WORK( I ) = S( I-N ) * WORK( I-N ) - AVG
! 267: END DO
! 268: CALL DLASSQ( N, WORK( N+1 ), 1, SCALE, SUMSQ )
! 269: STD = SCALE * SQRT( SUMSQ / N )
1.1 bertrand 270:
1.11 ! bertrand 271: IF ( STD .LT. TOL * AVG ) GOTO 999
1.1 bertrand 272:
1.11 ! bertrand 273: DO I = 1, N
! 274: T = ABS( A( I, I ) )
! 275: SI = S( I )
! 276: C2 = ( N-1 ) * T
! 277: C1 = ( N-2 ) * ( WORK( I ) - T*SI )
! 278: C0 = -(T*SI)*SI + 2*WORK( I )*SI - N*AVG
! 279: D = C1*C1 - 4*C0*C2
! 280:
! 281: IF ( D .LE. 0 ) THEN
! 282: INFO = -1
! 283: RETURN
! 284: END IF
! 285: SI = -2*C0 / ( C1 + SQRT( D ) )
! 286:
! 287: D = SI - S( I )
! 288: U = ZERO
! 289: IF ( UP ) THEN
! 290: DO J = 1, I
! 291: T = ABS( A( J, I ) )
! 292: U = U + S( J )*T
! 293: WORK( J ) = WORK( J ) + D*T
! 294: END DO
! 295: DO J = I+1,N
! 296: T = ABS( A( I, J ) )
! 297: U = U + S( J )*T
! 298: WORK( J ) = WORK( J ) + D*T
! 299: END DO
! 300: ELSE
! 301: DO J = 1, I
! 302: T = ABS( A( I, J ) )
! 303: U = U + S( J )*T
! 304: WORK( J ) = WORK( J ) + D*T
! 305: END DO
! 306: DO J = I+1,N
! 307: T = ABS( A( J, I ) )
! 308: U = U + S( J )*T
! 309: WORK( J ) = WORK( J ) + D*T
! 310: END DO
! 311: END IF
1.1 bertrand 312:
1.11 ! bertrand 313: AVG = AVG + ( U + WORK( I ) ) * D / N
! 314: S( I ) = SI
! 315: END DO
1.1 bertrand 316: END DO
317:
318: 999 CONTINUE
319:
320: SMLNUM = DLAMCH( 'SAFEMIN' )
321: BIGNUM = ONE / SMLNUM
322: SMIN = BIGNUM
323: SMAX = ZERO
1.11 ! bertrand 324: T = ONE / SQRT( AVG )
1.1 bertrand 325: BASE = DLAMCH( 'B' )
326: U = ONE / LOG( BASE )
327: DO I = 1, N
1.11 ! bertrand 328: S( I ) = BASE ** INT( U * LOG( S( I ) * T ) )
! 329: SMIN = MIN( SMIN, S( I ) )
! 330: SMAX = MAX( SMAX, S( I ) )
1.1 bertrand 331: END DO
332: SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
333: *
334: END
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