LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
spbequ.f
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1 *> \brief \b SPBEQU
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
22 *
23 * .. Scalar Arguments ..
24 * CHARACTER UPLO
25 * INTEGER INFO, KD, LDAB, N
26 * REAL AMAX, SCOND
27 * ..
28 * .. Array Arguments ..
29 * REAL AB( LDAB, * ), S( * )
30 * ..
31 *
32 *
33 *> \par Purpose:
34 * =============
35 *>
36 *> \verbatim
37 *>
38 *> SPBEQU 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 REAL 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 REAL 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 REAL
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 REAL
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 realOTHERcomputational
126 *
127 * =====================================================================
128  SUBROUTINE spbequ( 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  REAL AMAX, SCOND
138 * ..
139 * .. Array Arguments ..
140  REAL AB( LDAB, * ), S( * )
141 * ..
142 *
143 * =====================================================================
144 *
145 * .. Parameters ..
146  REAL ZERO, ONE
147  parameter( zero = 0.0e+0, one = 1.0e+0 )
148 * ..
149 * .. Local Scalars ..
150  LOGICAL UPPER
151  INTEGER I, J
152  REAL 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( 'SPBEQU', -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 SPBEQU
238 *
239  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine spbequ(UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO)
SPBEQU
Definition: spbequ.f:129