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
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