Annotation of rpl/lapack/lapack/zppequ.f, revision 1.14
1.8 bertrand 1: *> \brief \b ZPPEQU
2: *
3: * =========== DOCUMENTATION ===========
4: *
1.14 ! bertrand 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
1.8 bertrand 7: *
8: *> \htmlonly
1.14 ! bertrand 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">
1.8 bertrand 15: *> [TXT]</a>
1.14 ! bertrand 16: *> \endhtmlonly
1.8 bertrand 17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
1.14 ! bertrand 22: *
1.8 bertrand 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: * ..
1.14 ! bertrand 32: *
1.8 bertrand 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: *
1.14 ! bertrand 108: *> \author Univ. of Tennessee
! 109: *> \author Univ. of California Berkeley
! 110: *> \author Univ. of Colorado Denver
! 111: *> \author NAG Ltd.
1.8 bertrand 112: *
1.14 ! bertrand 113: *> \date December 2016
1.8 bertrand 114: *
115: *> \ingroup complex16OTHERcomputational
116: *
117: * =====================================================================
1.1 bertrand 118: SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
119: *
1.14 ! bertrand 120: * -- LAPACK computational routine (version 3.7.0) --
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..--
1.14 ! bertrand 123: * December 2016
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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