1: *> \brief \b ZPBEQU
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZPBEQU + dependencies
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11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zpbequ.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zpbequ.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, KD, LDAB, N
26: * DOUBLE PRECISION AMAX, SCOND
27: * ..
28: * .. Array Arguments ..
29: * DOUBLE PRECISION S( * )
30: * COMPLEX*16 AB( LDAB, * )
31: * ..
32: *
33: *
34: *> \par Purpose:
35: * =============
36: *>
37: *> \verbatim
38: *>
39: *> ZPBEQU computes row and column scalings intended to equilibrate a
40: *> Hermitian positive definite band matrix A and reduce its condition
41: *> number (with respect to the two-norm). S contains the scale factors,
42: *> S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
43: *> elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
44: *> choice of S puts the condition number of B within a factor N of the
45: *> 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 triangular of A is stored;
56: *> = 'L': Lower triangular 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] KD
66: *> \verbatim
67: *> KD is INTEGER
68: *> The number of superdiagonals of the matrix A if UPLO = 'U',
69: *> or the number of subdiagonals if UPLO = 'L'. KD >= 0.
70: *> \endverbatim
71: *>
72: *> \param[in] AB
73: *> \verbatim
74: *> AB is COMPLEX*16 array, dimension (LDAB,N)
75: *> The upper or lower triangle of the Hermitian band matrix A,
76: *> stored in the first KD+1 rows of the array. The j-th column
77: *> of A is stored in the j-th column of the array AB as follows:
78: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
79: *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
80: *> \endverbatim
81: *>
82: *> \param[in] LDAB
83: *> \verbatim
84: *> LDAB is INTEGER
85: *> The leading dimension of the array A. LDAB >= KD+1.
86: *> \endverbatim
87: *>
88: *> \param[out] S
89: *> \verbatim
90: *> S is DOUBLE PRECISION array, dimension (N)
91: *> If INFO = 0, S contains the scale factors for A.
92: *> \endverbatim
93: *>
94: *> \param[out] SCOND
95: *> \verbatim
96: *> SCOND is DOUBLE PRECISION
97: *> If INFO = 0, S contains the ratio of the smallest S(i) to
98: *> the largest S(i). If SCOND >= 0.1 and AMAX is neither too
99: *> large nor too small, it is not worth scaling by S.
100: *> \endverbatim
101: *>
102: *> \param[out] AMAX
103: *> \verbatim
104: *> AMAX is DOUBLE PRECISION
105: *> Absolute value of largest matrix element. If AMAX is very
106: *> close to overflow or very close to underflow, the matrix
107: *> should be scaled.
108: *> \endverbatim
109: *>
110: *> \param[out] INFO
111: *> \verbatim
112: *> INFO is INTEGER
113: *> = 0: successful exit
114: *> < 0: if INFO = -i, the i-th argument had an illegal value.
115: *> > 0: if INFO = i, the i-th diagonal element is nonpositive.
116: *> \endverbatim
117: *
118: * Authors:
119: * ========
120: *
121: *> \author Univ. of Tennessee
122: *> \author Univ. of California Berkeley
123: *> \author Univ. of Colorado Denver
124: *> \author NAG Ltd.
125: *
126: *> \date November 2011
127: *
128: *> \ingroup complex16OTHERcomputational
129: *
130: * =====================================================================
131: SUBROUTINE ZPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
132: *
133: * -- LAPACK computational routine (version 3.4.0) --
134: * -- LAPACK is a software package provided by Univ. of Tennessee, --
135: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
136: * November 2011
137: *
138: * .. Scalar Arguments ..
139: CHARACTER UPLO
140: INTEGER INFO, KD, LDAB, N
141: DOUBLE PRECISION AMAX, SCOND
142: * ..
143: * .. Array Arguments ..
144: DOUBLE PRECISION S( * )
145: COMPLEX*16 AB( LDAB, * )
146: * ..
147: *
148: * =====================================================================
149: *
150: * .. Parameters ..
151: DOUBLE PRECISION ZERO, ONE
152: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
153: * ..
154: * .. Local Scalars ..
155: LOGICAL UPPER
156: INTEGER I, J
157: DOUBLE PRECISION SMIN
158: * ..
159: * .. External Functions ..
160: LOGICAL LSAME
161: EXTERNAL LSAME
162: * ..
163: * .. External Subroutines ..
164: EXTERNAL XERBLA
165: * ..
166: * .. Intrinsic Functions ..
167: INTRINSIC DBLE, MAX, MIN, SQRT
168: * ..
169: * .. Executable Statements ..
170: *
171: * Test the input parameters.
172: *
173: INFO = 0
174: UPPER = LSAME( UPLO, 'U' )
175: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
176: INFO = -1
177: ELSE IF( N.LT.0 ) THEN
178: INFO = -2
179: ELSE IF( KD.LT.0 ) THEN
180: INFO = -3
181: ELSE IF( LDAB.LT.KD+1 ) THEN
182: INFO = -5
183: END IF
184: IF( INFO.NE.0 ) THEN
185: CALL XERBLA( 'ZPBEQU', -INFO )
186: RETURN
187: END IF
188: *
189: * Quick return if possible
190: *
191: IF( N.EQ.0 ) THEN
192: SCOND = ONE
193: AMAX = ZERO
194: RETURN
195: END IF
196: *
197: IF( UPPER ) THEN
198: J = KD + 1
199: ELSE
200: J = 1
201: END IF
202: *
203: * Initialize SMIN and AMAX.
204: *
205: S( 1 ) = DBLE( AB( J, 1 ) )
206: SMIN = S( 1 )
207: AMAX = S( 1 )
208: *
209: * Find the minimum and maximum diagonal elements.
210: *
211: DO 10 I = 2, N
212: S( I ) = DBLE( AB( J, I ) )
213: SMIN = MIN( SMIN, S( I ) )
214: AMAX = MAX( AMAX, S( I ) )
215: 10 CONTINUE
216: *
217: IF( SMIN.LE.ZERO ) THEN
218: *
219: * Find the first non-positive diagonal element and return.
220: *
221: DO 20 I = 1, N
222: IF( S( I ).LE.ZERO ) THEN
223: INFO = I
224: RETURN
225: END IF
226: 20 CONTINUE
227: ELSE
228: *
229: * Set the scale factors to the reciprocals
230: * of the diagonal elements.
231: *
232: DO 30 I = 1, N
233: S( I ) = ONE / SQRT( S( I ) )
234: 30 CONTINUE
235: *
236: * Compute SCOND = min(S(I)) / max(S(I))
237: *
238: SCOND = SQRT( SMIN ) / SQRT( AMAX )
239: END IF
240: RETURN
241: *
242: * End of ZPBEQU
243: *
244: END
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