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: *> \ingroup complex16OTHERcomputational
114: *
115: * =====================================================================
116: SUBROUTINE ZPPEQU( UPLO, N, AP, S, SCOND, AMAX, INFO )
117: *
118: * -- LAPACK computational routine --
119: * -- LAPACK is a software package provided by Univ. of Tennessee, --
120: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
121: *
122: * .. Scalar Arguments ..
123: CHARACTER UPLO
124: INTEGER INFO, N
125: DOUBLE PRECISION AMAX, SCOND
126: * ..
127: * .. Array Arguments ..
128: DOUBLE PRECISION S( * )
129: COMPLEX*16 AP( * )
130: * ..
131: *
132: * =====================================================================
133: *
134: * .. Parameters ..
135: DOUBLE PRECISION ONE, ZERO
136: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
137: * ..
138: * .. Local Scalars ..
139: LOGICAL UPPER
140: INTEGER I, JJ
141: DOUBLE PRECISION SMIN
142: * ..
143: * .. External Functions ..
144: LOGICAL LSAME
145: EXTERNAL LSAME
146: * ..
147: * .. External Subroutines ..
148: EXTERNAL XERBLA
149: * ..
150: * .. Intrinsic Functions ..
151: INTRINSIC DBLE, MAX, MIN, SQRT
152: * ..
153: * .. Executable Statements ..
154: *
155: * Test the input parameters.
156: *
157: INFO = 0
158: UPPER = LSAME( UPLO, 'U' )
159: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
160: INFO = -1
161: ELSE IF( N.LT.0 ) THEN
162: INFO = -2
163: END IF
164: IF( INFO.NE.0 ) THEN
165: CALL XERBLA( 'ZPPEQU', -INFO )
166: RETURN
167: END IF
168: *
169: * Quick return if possible
170: *
171: IF( N.EQ.0 ) THEN
172: SCOND = ONE
173: AMAX = ZERO
174: RETURN
175: END IF
176: *
177: * Initialize SMIN and AMAX.
178: *
179: S( 1 ) = DBLE( AP( 1 ) )
180: SMIN = S( 1 )
181: AMAX = S( 1 )
182: *
183: IF( UPPER ) THEN
184: *
185: * UPLO = 'U': Upper triangle of A is stored.
186: * Find the minimum and maximum diagonal elements.
187: *
188: JJ = 1
189: DO 10 I = 2, N
190: JJ = JJ + I
191: S( I ) = DBLE( AP( JJ ) )
192: SMIN = MIN( SMIN, S( I ) )
193: AMAX = MAX( AMAX, S( I ) )
194: 10 CONTINUE
195: *
196: ELSE
197: *
198: * UPLO = 'L': Lower triangle of A is stored.
199: * Find the minimum and maximum diagonal elements.
200: *
201: JJ = 1
202: DO 20 I = 2, N
203: JJ = JJ + N - I + 2
204: S( I ) = DBLE( AP( JJ ) )
205: SMIN = MIN( SMIN, S( I ) )
206: AMAX = MAX( AMAX, S( I ) )
207: 20 CONTINUE
208: END IF
209: *
210: IF( SMIN.LE.ZERO ) THEN
211: *
212: * Find the first non-positive diagonal element and return.
213: *
214: DO 30 I = 1, N
215: IF( S( I ).LE.ZERO ) THEN
216: INFO = I
217: RETURN
218: END IF
219: 30 CONTINUE
220: ELSE
221: *
222: * Set the scale factors to the reciprocals
223: * of the diagonal elements.
224: *
225: DO 40 I = 1, N
226: S( I ) = ONE / SQRT( S( I ) )
227: 40 CONTINUE
228: *
229: * Compute SCOND = min(S(I)) / max(S(I))
230: *
231: SCOND = SQRT( SMIN ) / SQRT( AMAX )
232: END IF
233: RETURN
234: *
235: * End of ZPPEQU
236: *
237: END
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