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