Annotation of rpl/lapack/lapack/dpbequ.f, revision 1.2
1.1 bertrand 1: SUBROUTINE DPBEQU( UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, INFO )
2: *
3: * -- LAPACK routine (version 3.2) --
4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
6: * November 2006
7: *
8: * .. Scalar Arguments ..
9: CHARACTER UPLO
10: INTEGER INFO, KD, LDAB, N
11: DOUBLE PRECISION AMAX, SCOND
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION AB( LDAB, * ), S( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DPBEQU computes row and column scalings intended to equilibrate a
21: * symmetric positive definite band matrix A and reduce its condition
22: * number (with respect to the two-norm). S contains the scale factors,
23: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
24: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
25: * choice of S puts the condition number of B within a factor N of the
26: * smallest possible condition number over all possible diagonal
27: * scalings.
28: *
29: * Arguments
30: * =========
31: *
32: * UPLO (input) CHARACTER*1
33: * = 'U': Upper triangular of A is stored;
34: * = 'L': Lower triangular of A is stored.
35: *
36: * N (input) INTEGER
37: * The order of the matrix A. N >= 0.
38: *
39: * KD (input) INTEGER
40: * The number of superdiagonals of the matrix A if UPLO = 'U',
41: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
42: *
43: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
44: * The upper or lower triangle of the symmetric band matrix A,
45: * stored in the first KD+1 rows of the array. The j-th column
46: * of A is stored in the j-th column of the array AB as follows:
47: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
48: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
49: *
50: * LDAB (input) INTEGER
51: * The leading dimension of the array A. LDAB >= KD+1.
52: *
53: * S (output) DOUBLE PRECISION array, dimension (N)
54: * If INFO = 0, S contains the scale factors for A.
55: *
56: * SCOND (output) DOUBLE PRECISION
57: * If INFO = 0, S contains the ratio of the smallest S(i) to
58: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
59: * large nor too small, it is not worth scaling by S.
60: *
61: * AMAX (output) DOUBLE PRECISION
62: * Absolute value of largest matrix element. If AMAX is very
63: * close to overflow or very close to underflow, the matrix
64: * should be scaled.
65: *
66: * INFO (output) INTEGER
67: * = 0: successful exit
68: * < 0: if INFO = -i, the i-th argument had an illegal value.
69: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
70: *
71: * =====================================================================
72: *
73: * .. Parameters ..
74: DOUBLE PRECISION ZERO, ONE
75: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
76: * ..
77: * .. Local Scalars ..
78: LOGICAL UPPER
79: INTEGER I, J
80: DOUBLE PRECISION SMIN
81: * ..
82: * .. External Functions ..
83: LOGICAL LSAME
84: EXTERNAL LSAME
85: * ..
86: * .. External Subroutines ..
87: EXTERNAL XERBLA
88: * ..
89: * .. Intrinsic Functions ..
90: INTRINSIC MAX, MIN, SQRT
91: * ..
92: * .. Executable Statements ..
93: *
94: * Test the input parameters.
95: *
96: INFO = 0
97: UPPER = LSAME( UPLO, 'U' )
98: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
99: INFO = -1
100: ELSE IF( N.LT.0 ) THEN
101: INFO = -2
102: ELSE IF( KD.LT.0 ) THEN
103: INFO = -3
104: ELSE IF( LDAB.LT.KD+1 ) THEN
105: INFO = -5
106: END IF
107: IF( INFO.NE.0 ) THEN
108: CALL XERBLA( 'DPBEQU', -INFO )
109: RETURN
110: END IF
111: *
112: * Quick return if possible
113: *
114: IF( N.EQ.0 ) THEN
115: SCOND = ONE
116: AMAX = ZERO
117: RETURN
118: END IF
119: *
120: IF( UPPER ) THEN
121: J = KD + 1
122: ELSE
123: J = 1
124: END IF
125: *
126: * Initialize SMIN and AMAX.
127: *
128: S( 1 ) = AB( J, 1 )
129: SMIN = S( 1 )
130: AMAX = S( 1 )
131: *
132: * Find the minimum and maximum diagonal elements.
133: *
134: DO 10 I = 2, N
135: S( I ) = AB( J, I )
136: SMIN = MIN( SMIN, S( I ) )
137: AMAX = MAX( AMAX, S( I ) )
138: 10 CONTINUE
139: *
140: IF( SMIN.LE.ZERO ) THEN
141: *
142: * Find the first non-positive diagonal element and return.
143: *
144: DO 20 I = 1, N
145: IF( S( I ).LE.ZERO ) THEN
146: INFO = I
147: RETURN
148: END IF
149: 20 CONTINUE
150: ELSE
151: *
152: * Set the scale factors to the reciprocals
153: * of the diagonal elements.
154: *
155: DO 30 I = 1, N
156: S( I ) = ONE / SQRT( S( I ) )
157: 30 CONTINUE
158: *
159: * Compute SCOND = min(S(I)) / max(S(I))
160: *
161: SCOND = SQRT( SMIN ) / SQRT( AMAX )
162: END IF
163: RETURN
164: *
165: * End of DPBEQU
166: *
167: END
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