Annotation of rpl/lapack/lapack/dppequ.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b DPPEQU
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DPPEQU + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dppequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dppequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dppequ.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * CHARACTER UPLO
! 25: * INTEGER INFO, N
! 26: * DOUBLE PRECISION AMAX, SCOND
! 27: * ..
! 28: * .. Array Arguments ..
! 29: * DOUBLE PRECISION AP( * ), S( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DPPEQU computes row and column scalings intended to equilibrate a
! 39: *> symmetric positive definite matrix A in packed storage and reduce
! 40: *> its condition number (with respect to the two-norm). S contains the
! 41: *> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
! 42: *> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
! 43: *> This choice of S puts the condition number of B 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] AP
! 65: *> \verbatim
! 66: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 67: *> The upper or lower triangle of the symmetric matrix A, packed
! 68: *> columnwise in a linear array. The j-th column of A is stored
! 69: *> in the array AP as follows:
! 70: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 71: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 72: *> \endverbatim
! 73: *>
! 74: *> \param[out] S
! 75: *> \verbatim
! 76: *> S is DOUBLE PRECISION array, dimension (N)
! 77: *> If INFO = 0, S contains the scale factors for A.
! 78: *> \endverbatim
! 79: *>
! 80: *> \param[out] SCOND
! 81: *> \verbatim
! 82: *> SCOND is DOUBLE PRECISION
! 83: *> If INFO = 0, S contains the ratio of the smallest S(i) to
! 84: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 85: *> large nor too small, it is not worth scaling by S.
! 86: *> \endverbatim
! 87: *>
! 88: *> \param[out] AMAX
! 89: *> \verbatim
! 90: *> AMAX is DOUBLE PRECISION
! 91: *> Absolute value of largest matrix element. If AMAX is very
! 92: *> close to overflow or very close to underflow, the matrix
! 93: *> should be scaled.
! 94: *> \endverbatim
! 95: *>
! 96: *> \param[out] INFO
! 97: *> \verbatim
! 98: *> INFO is INTEGER
! 99: *> = 0: successful exit
! 100: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 101: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 102: *> \endverbatim
! 103: *
! 104: * Authors:
! 105: * ========
! 106: *
! 107: *> \author Univ. of Tennessee
! 108: *> \author Univ. of California Berkeley
! 109: *> \author Univ. of Colorado Denver
! 110: *> \author NAG Ltd.
! 111: *
! 112: *> \date November 2011
! 113: *
! 114: *> \ingroup doubleOTHERcomputational
! 115: *
! 116: * =====================================================================
1.1 bertrand 117: SUBROUTINE DPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
118: *
1.8 ! bertrand 119: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 120: * -- LAPACK is a software package provided by Univ. of Tennessee, --
121: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 122: * November 2011
1.1 bertrand 123: *
124: * .. Scalar Arguments ..
125: CHARACTER UPLO
126: INTEGER INFO, N
127: DOUBLE PRECISION AMAX, SCOND
128: * ..
129: * .. Array Arguments ..
130: DOUBLE PRECISION AP( * ), S( * )
131: * ..
132: *
133: * =====================================================================
134: *
135: * .. Parameters ..
136: DOUBLE PRECISION ONE, ZERO
137: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
138: * ..
139: * .. Local Scalars ..
140: LOGICAL UPPER
141: INTEGER I, JJ
142: DOUBLE PRECISION SMIN
143: * ..
144: * .. External Functions ..
145: LOGICAL LSAME
146: EXTERNAL LSAME
147: * ..
148: * .. External Subroutines ..
149: EXTERNAL XERBLA
150: * ..
151: * .. Intrinsic Functions ..
152: INTRINSIC MAX, MIN, SQRT
153: * ..
154: * .. Executable Statements ..
155: *
156: * Test the input parameters.
157: *
158: INFO = 0
159: UPPER = LSAME( UPLO, 'U' )
160: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
161: INFO = -1
162: ELSE IF( N.LT.0 ) THEN
163: INFO = -2
164: END IF
165: IF( INFO.NE.0 ) THEN
166: CALL XERBLA( 'DPPEQU', -INFO )
167: RETURN
168: END IF
169: *
170: * Quick return if possible
171: *
172: IF( N.EQ.0 ) THEN
173: SCOND = ONE
174: AMAX = ZERO
175: RETURN
176: END IF
177: *
178: * Initialize SMIN and AMAX.
179: *
180: S( 1 ) = AP( 1 )
181: SMIN = S( 1 )
182: AMAX = S( 1 )
183: *
184: IF( UPPER ) THEN
185: *
186: * UPLO = 'U': Upper triangle of A is stored.
187: * Find the minimum and maximum diagonal elements.
188: *
189: JJ = 1
190: DO 10 I = 2, N
191: JJ = JJ + I
192: S( I ) = AP( JJ )
193: SMIN = MIN( SMIN, S( I ) )
194: AMAX = MAX( AMAX, S( I ) )
195: 10 CONTINUE
196: *
197: ELSE
198: *
199: * UPLO = 'L': Lower triangle of A is stored.
200: * Find the minimum and maximum diagonal elements.
201: *
202: JJ = 1
203: DO 20 I = 2, N
204: JJ = JJ + N - I + 2
205: S( I ) = AP( JJ )
206: SMIN = MIN( SMIN, S( I ) )
207: AMAX = MAX( AMAX, S( I ) )
208: 20 CONTINUE
209: END IF
210: *
211: IF( SMIN.LE.ZERO ) THEN
212: *
213: * Find the first non-positive diagonal element and return.
214: *
215: DO 30 I = 1, N
216: IF( S( I ).LE.ZERO ) THEN
217: INFO = I
218: RETURN
219: END IF
220: 30 CONTINUE
221: ELSE
222: *
223: * Set the scale factors to the reciprocals
224: * of the diagonal elements.
225: *
226: DO 40 I = 1, N
227: S( I ) = ONE / SQRT( S( I ) )
228: 40 CONTINUE
229: *
230: * Compute SCOND = min(S(I)) / max(S(I))
231: *
232: SCOND = SQRT( SMIN ) / SQRT( AMAX )
233: END IF
234: RETURN
235: *
236: * End of DPPEQU
237: *
238: END
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