Annotation of rpl/lapack/lapack/zppequ.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b ZPPEQU
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZPPEQU + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zppequ.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zppequ.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zppequ.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZPPEQU( 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 S( * )
! 30: * COMPLEX*16 AP( * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *>
! 39: *> ZPPEQU computes row and column scalings intended to equilibrate a
! 40: *> Hermitian positive definite matrix A in packed storage and reduce
! 41: *> its condition number (with respect to the two-norm). S contains the
! 42: *> scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
! 43: *> B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
! 44: *> This choice of S puts the condition number of B within a factor N of
! 45: *> the smallest possible condition number over all possible diagonal
! 46: *> scalings.
! 47: *> \endverbatim
! 48: *
! 49: * Arguments:
! 50: * ==========
! 51: *
! 52: *> \param[in] UPLO
! 53: *> \verbatim
! 54: *> UPLO is CHARACTER*1
! 55: *> = 'U': Upper triangle of A is stored;
! 56: *> = 'L': Lower triangle of A is stored.
! 57: *> \endverbatim
! 58: *>
! 59: *> \param[in] N
! 60: *> \verbatim
! 61: *> N is INTEGER
! 62: *> The order of the matrix A. N >= 0.
! 63: *> \endverbatim
! 64: *>
! 65: *> \param[in] AP
! 66: *> \verbatim
! 67: *> AP is COMPLEX*16 array, dimension (N*(N+1)/2)
! 68: *> The upper or lower triangle of the Hermitian matrix A, packed
! 69: *> columnwise in a linear array. The j-th column of A is stored
! 70: *> in the array AP as follows:
! 71: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 72: *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[out] S
! 76: *> \verbatim
! 77: *> S is DOUBLE PRECISION array, dimension (N)
! 78: *> If INFO = 0, S contains the scale factors for A.
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[out] SCOND
! 82: *> \verbatim
! 83: *> SCOND is DOUBLE PRECISION
! 84: *> If INFO = 0, S contains the ratio of the smallest S(i) to
! 85: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
! 86: *> large nor too small, it is not worth scaling by S.
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[out] AMAX
! 90: *> \verbatim
! 91: *> AMAX is DOUBLE PRECISION
! 92: *> Absolute value of largest matrix element. If AMAX is very
! 93: *> close to overflow or very close to underflow, the matrix
! 94: *> should be scaled.
! 95: *> \endverbatim
! 96: *>
! 97: *> \param[out] INFO
! 98: *> \verbatim
! 99: *> INFO is INTEGER
! 100: *> = 0: successful exit
! 101: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 102: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
! 103: *> \endverbatim
! 104: *
! 105: * Authors:
! 106: * ========
! 107: *
! 108: *> \author Univ. of Tennessee
! 109: *> \author Univ. of California Berkeley
! 110: *> \author Univ. of Colorado Denver
! 111: *> \author NAG Ltd.
! 112: *
! 113: *> \date November 2011
! 114: *
! 115: *> \ingroup complex16OTHERcomputational
! 116: *
! 117: * =====================================================================
1.1 bertrand 118: SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
119: *
1.8 ! bertrand 120: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 121: * -- LAPACK is a software package provided by Univ. of Tennessee, --
122: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 123: * November 2011
1.1 bertrand 124: *
125: * .. Scalar Arguments ..
126: CHARACTER UPLO
127: INTEGER INFO, N
128: DOUBLE PRECISION AMAX, SCOND
129: * ..
130: * .. Array Arguments ..
131: DOUBLE PRECISION S( * )
132: COMPLEX*16 AP( * )
133: * ..
134: *
135: * =====================================================================
136: *
137: * .. Parameters ..
138: DOUBLE PRECISION ONE, ZERO
139: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
140: * ..
141: * .. Local Scalars ..
142: LOGICAL UPPER
143: INTEGER I, JJ
144: DOUBLE PRECISION SMIN
145: * ..
146: * .. External Functions ..
147: LOGICAL LSAME
148: EXTERNAL LSAME
149: * ..
150: * .. External Subroutines ..
151: EXTERNAL XERBLA
152: * ..
153: * .. Intrinsic Functions ..
154: INTRINSIC DBLE, MAX, MIN, SQRT
155: * ..
156: * .. Executable Statements ..
157: *
158: * Test the input parameters.
159: *
160: INFO = 0
161: UPPER = LSAME( UPLO, 'U' )
162: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
163: INFO = -1
164: ELSE IF( N.LT.0 ) THEN
165: INFO = -2
166: END IF
167: IF( INFO.NE.0 ) THEN
168: CALL XERBLA( 'ZPPEQU', -INFO )
169: RETURN
170: END IF
171: *
172: * Quick return if possible
173: *
174: IF( N.EQ.0 ) THEN
175: SCOND = ONE
176: AMAX = ZERO
177: RETURN
178: END IF
179: *
180: * Initialize SMIN and AMAX.
181: *
182: S( 1 ) = DBLE( AP( 1 ) )
183: SMIN = S( 1 )
184: AMAX = S( 1 )
185: *
186: IF( UPPER ) THEN
187: *
188: * UPLO = 'U': Upper triangle of A is stored.
189: * Find the minimum and maximum diagonal elements.
190: *
191: JJ = 1
192: DO 10 I = 2, N
193: JJ = JJ + I
194: S( I ) = DBLE( AP( JJ ) )
195: SMIN = MIN( SMIN, S( I ) )
196: AMAX = MAX( AMAX, S( I ) )
197: 10 CONTINUE
198: *
199: ELSE
200: *
201: * UPLO = 'L': Lower triangle of A is stored.
202: * Find the minimum and maximum diagonal elements.
203: *
204: JJ = 1
205: DO 20 I = 2, N
206: JJ = JJ + N - I + 2
207: S( I ) = DBLE( AP( JJ ) )
208: SMIN = MIN( SMIN, S( I ) )
209: AMAX = MAX( AMAX, S( I ) )
210: 20 CONTINUE
211: END IF
212: *
213: IF( SMIN.LE.ZERO ) THEN
214: *
215: * Find the first non-positive diagonal element and return.
216: *
217: DO 30 I = 1, N
218: IF( S( I ).LE.ZERO ) THEN
219: INFO = I
220: RETURN
221: END IF
222: 30 CONTINUE
223: ELSE
224: *
225: * Set the scale factors to the reciprocals
226: * of the diagonal elements.
227: *
228: DO 40 I = 1, N
229: S( I ) = ONE / SQRT( S( I ) )
230: 40 CONTINUE
231: *
232: * Compute SCOND = min(S(I)) / max(S(I))
233: *
234: SCOND = SQRT( SMIN ) / SQRT( AMAX )
235: END IF
236: RETURN
237: *
238: * End of ZPPEQU
239: *
240: END
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