Annotation of rpl/lapack/lapack/zpbequ.f, revision 1.4
1.1 bertrand 1: SUBROUTINE ZPBEQU( 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 S( * )
15: COMPLEX*16 AB( LDAB, * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * ZPBEQU computes row and column scalings intended to equilibrate a
22: * Hermitian positive definite band matrix A and reduce its condition
23: * number (with respect to the two-norm). S contains the scale factors,
24: * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with
25: * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This
26: * choice of S puts the condition number of B within a factor N of the
27: * smallest possible condition number over all possible diagonal
28: * scalings.
29: *
30: * Arguments
31: * =========
32: *
33: * UPLO (input) CHARACTER*1
34: * = 'U': Upper triangular of A is stored;
35: * = 'L': Lower triangular of A is stored.
36: *
37: * N (input) INTEGER
38: * The order of the matrix A. N >= 0.
39: *
40: * KD (input) INTEGER
41: * The number of superdiagonals of the matrix A if UPLO = 'U',
42: * or the number of subdiagonals if UPLO = 'L'. KD >= 0.
43: *
44: * AB (input) COMPLEX*16 array, dimension (LDAB,N)
45: * The upper or lower triangle of the Hermitian band matrix A,
46: * stored in the first KD+1 rows of the array. The j-th column
47: * of A is stored in the j-th column of the array AB as follows:
48: * if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
49: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
50: *
51: * LDAB (input) INTEGER
52: * The leading dimension of the array A. LDAB >= KD+1.
53: *
54: * S (output) DOUBLE PRECISION array, dimension (N)
55: * If INFO = 0, S contains the scale factors for A.
56: *
57: * SCOND (output) DOUBLE PRECISION
58: * If INFO = 0, S contains the ratio of the smallest S(i) to
59: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
60: * large nor too small, it is not worth scaling by S.
61: *
62: * AMAX (output) DOUBLE PRECISION
63: * Absolute value of largest matrix element. If AMAX is very
64: * close to overflow or very close to underflow, the matrix
65: * should be scaled.
66: *
67: * INFO (output) INTEGER
68: * = 0: successful exit
69: * < 0: if INFO = -i, the i-th argument had an illegal value.
70: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
71: *
72: * =====================================================================
73: *
74: * .. Parameters ..
75: DOUBLE PRECISION ZERO, ONE
76: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
77: * ..
78: * .. Local Scalars ..
79: LOGICAL UPPER
80: INTEGER I, J
81: DOUBLE PRECISION SMIN
82: * ..
83: * .. External Functions ..
84: LOGICAL LSAME
85: EXTERNAL LSAME
86: * ..
87: * .. External Subroutines ..
88: EXTERNAL XERBLA
89: * ..
90: * .. Intrinsic Functions ..
91: INTRINSIC DBLE, MAX, MIN, SQRT
92: * ..
93: * .. Executable Statements ..
94: *
95: * Test the input parameters.
96: *
97: INFO = 0
98: UPPER = LSAME( UPLO, 'U' )
99: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
100: INFO = -1
101: ELSE IF( N.LT.0 ) THEN
102: INFO = -2
103: ELSE IF( KD.LT.0 ) THEN
104: INFO = -3
105: ELSE IF( LDAB.LT.KD+1 ) THEN
106: INFO = -5
107: END IF
108: IF( INFO.NE.0 ) THEN
109: CALL XERBLA( 'ZPBEQU', -INFO )
110: RETURN
111: END IF
112: *
113: * Quick return if possible
114: *
115: IF( N.EQ.0 ) THEN
116: SCOND = ONE
117: AMAX = ZERO
118: RETURN
119: END IF
120: *
121: IF( UPPER ) THEN
122: J = KD + 1
123: ELSE
124: J = 1
125: END IF
126: *
127: * Initialize SMIN and AMAX.
128: *
129: S( 1 ) = DBLE( AB( J, 1 ) )
130: SMIN = S( 1 )
131: AMAX = S( 1 )
132: *
133: * Find the minimum and maximum diagonal elements.
134: *
135: DO 10 I = 2, N
136: S( I ) = DBLE( AB( J, I ) )
137: SMIN = MIN( SMIN, S( I ) )
138: AMAX = MAX( AMAX, S( I ) )
139: 10 CONTINUE
140: *
141: IF( SMIN.LE.ZERO ) THEN
142: *
143: * Find the first non-positive diagonal element and return.
144: *
145: DO 20 I = 1, N
146: IF( S( I ).LE.ZERO ) THEN
147: INFO = I
148: RETURN
149: END IF
150: 20 CONTINUE
151: ELSE
152: *
153: * Set the scale factors to the reciprocals
154: * of the diagonal elements.
155: *
156: DO 30 I = 1, N
157: S( I ) = ONE / SQRT( S( I ) )
158: 30 CONTINUE
159: *
160: * Compute SCOND = min(S(I)) / max(S(I))
161: *
162: SCOND = SQRT( SMIN ) / SQRT( AMAX )
163: END IF
164: RETURN
165: *
166: * End of ZPBEQU
167: *
168: END
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