Annotation of rpl/lapack/lapack/zpoequb.f, revision 1.6
1.6 ! bertrand 1: *> \brief \b ZPOEQUB
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZPOEQUB + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zpoequb.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpoequb.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpoequb.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * INTEGER INFO, LDA, N
! 25: * DOUBLE PRECISION AMAX, SCOND
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * COMPLEX*16 A( LDA, * )
! 29: * DOUBLE PRECISION S( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZPOEQUB computes row and column scalings intended to equilibrate a
! 39: *> symmetric positive definite matrix A and reduce its condition number
! 40: *> (with respect to the two-norm). S contains the scale factors,
! 41: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
! 42: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
! 43: *> choice of S puts the condition number of B within a factor N of the
! 44: *> smallest possible condition number over all possible diagonal
! 45: *> scalings.
! 46: *> \endverbatim
! 47: *
! 48: * Arguments:
! 49: * ==========
! 50: *
! 51: *> \param[in] N
! 52: *> \verbatim
! 53: *> N is INTEGER
! 54: *> The order of the matrix A. N >= 0.
! 55: *> \endverbatim
! 56: *>
! 57: *> \param[in] A
! 58: *> \verbatim
! 59: *> A is COMPLEX*16 array, dimension (LDA,N)
! 60: *> The N-by-N symmetric positive definite matrix whose scaling
! 61: *> factors are to be computed. Only the diagonal elements of A
! 62: *> are referenced.
! 63: *> \endverbatim
! 64: *>
! 65: *> \param[in] LDA
! 66: *> \verbatim
! 67: *> LDA is INTEGER
! 68: *> The leading dimension of the array A. LDA >= max(1,N).
! 69: *> \endverbatim
! 70: *>
! 71: *> \param[out] S
! 72: *> \verbatim
! 73: *> S is DOUBLE PRECISION array, dimension (N)
! 74: *> If INFO = 0, S contains the scale factors for A.
! 75: *> \endverbatim
! 76: *>
! 77: *> \param[out] SCOND
! 78: *> \verbatim
! 79: *> SCOND is DOUBLE PRECISION
! 80: *> If INFO = 0, S contains the ratio of the smallest S(i) to
! 81: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 82: *> large nor too small, it is not worth scaling by S.
! 83: *> \endverbatim
! 84: *>
! 85: *> \param[out] AMAX
! 86: *> \verbatim
! 87: *> AMAX is DOUBLE PRECISION
! 88: *> Absolute value of largest matrix element. If AMAX is very
! 89: *> close to overflow or very close to underflow, the matrix
! 90: *> should be scaled.
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[out] INFO
! 94: *> \verbatim
! 95: *> INFO is INTEGER
! 96: *> = 0: successful exit
! 97: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 98: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 99: *> \endverbatim
! 100: *
! 101: * Authors:
! 102: * ========
! 103: *
! 104: *> \author Univ. of Tennessee
! 105: *> \author Univ. of California Berkeley
! 106: *> \author Univ. of Colorado Denver
! 107: *> \author NAG Ltd.
! 108: *
! 109: *> \date November 2011
! 110: *
! 111: *> \ingroup complex16POcomputational
! 112: *
! 113: * =====================================================================
1.1 bertrand 114: SUBROUTINE ZPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
115: *
1.6 ! bertrand 116: * -- LAPACK computational routine (version 3.4.0) --
! 117: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 118: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 119: * November 2011
1.1 bertrand 120: *
121: * .. Scalar Arguments ..
122: INTEGER INFO, LDA, N
123: DOUBLE PRECISION AMAX, SCOND
124: * ..
125: * .. Array Arguments ..
126: COMPLEX*16 A( LDA, * )
127: DOUBLE PRECISION S( * )
128: * ..
129: *
130: * =====================================================================
131: *
132: * .. Parameters ..
133: DOUBLE PRECISION ZERO, ONE
134: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
135: * ..
136: * .. Local Scalars ..
137: INTEGER I
138: DOUBLE PRECISION SMIN, BASE, TMP
139: * ..
140: * .. External Functions ..
141: DOUBLE PRECISION DLAMCH
142: EXTERNAL DLAMCH
143: * ..
144: * .. External Subroutines ..
145: EXTERNAL XERBLA
146: * ..
147: * .. Intrinsic Functions ..
148: INTRINSIC MAX, MIN, SQRT, LOG, INT, REAL, DIMAG
149: * ..
150: * .. Executable Statements ..
151: *
152: * Test the input parameters.
153: *
154: * Positive definite only performs 1 pass of equilibration.
155: *
156: INFO = 0
157: IF( N.LT.0 ) THEN
158: INFO = -1
159: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
160: INFO = -3
161: END IF
162: IF( INFO.NE.0 ) THEN
163: CALL XERBLA( 'ZPOEQUB', -INFO )
164: RETURN
165: END IF
166: *
167: * Quick return if possible.
168: *
169: IF( N.EQ.0 ) THEN
170: SCOND = ONE
171: AMAX = ZERO
172: RETURN
173: END IF
174:
175: BASE = DLAMCH( 'B' )
176: TMP = -0.5D+0 / LOG ( BASE )
177: *
178: * Find the minimum and maximum diagonal elements.
179: *
180: S( 1 ) = A( 1, 1 )
181: SMIN = S( 1 )
182: AMAX = S( 1 )
183: DO 10 I = 2, N
184: S( I ) = A( I, I )
185: SMIN = MIN( SMIN, S( I ) )
186: AMAX = MAX( AMAX, S( I ) )
187: 10 CONTINUE
188: *
189: IF( SMIN.LE.ZERO ) THEN
190: *
191: * Find the first non-positive diagonal element and return.
192: *
193: DO 20 I = 1, N
194: IF( S( I ).LE.ZERO ) THEN
195: INFO = I
196: RETURN
197: END IF
198: 20 CONTINUE
199: ELSE
200: *
201: * Set the scale factors to the reciprocals
202: * of the diagonal elements.
203: *
204: DO 30 I = 1, N
205: S( I ) = BASE ** INT( TMP * LOG( S( I ) ) )
206: 30 CONTINUE
207: *
208: * Compute SCOND = min(S(I)) / max(S(I)).
209: *
210: SCOND = SQRT( SMIN ) / SQRT( AMAX )
211: END IF
212: *
213: RETURN
214: *
215: * End of ZPOEQUB
216: *
217: END
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