Annotation of rpl/lapack/lapack/zppequ.f, revision 1.7
1.1 bertrand 1: SUBROUTINE ZPPEQU( UPLO, N, AP, 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, N
11: DOUBLE PRECISION AMAX, SCOND
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION S( * )
15: COMPLEX*16 AP( * )
16: * ..
17: *
18: * Purpose
19: * =======
20: *
21: * ZPPEQU computes row and column scalings intended to equilibrate a
22: * Hermitian positive definite matrix A in packed storage and reduce
23: * its condition number (with respect to the two-norm). S contains the
24: * scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
25: * B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
26: * This choice of S puts the condition number of B within a factor N of
27: * the smallest possible condition number over all possible diagonal
28: * scalings.
29: *
30: * Arguments
31: * =========
32: *
33: * UPLO (input) CHARACTER*1
34: * = 'U': Upper triangle of A is stored;
35: * = 'L': Lower triangle of A is stored.
36: *
37: * N (input) INTEGER
38: * The order of the matrix A. N >= 0.
39: *
40: * AP (input) COMPLEX*16 array, dimension (N*(N+1)/2)
41: * The upper or lower triangle of the Hermitian matrix A, packed
42: * columnwise in a linear array. The j-th column of A is stored
43: * in the array AP as follows:
44: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
45: * if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
46: *
47: * S (output) DOUBLE PRECISION array, dimension (N)
48: * If INFO = 0, S contains the scale factors for A.
49: *
50: * SCOND (output) DOUBLE PRECISION
51: * If INFO = 0, S contains the ratio of the smallest S(i) to
52: * the largest S(i). If SCOND >= 0.1 and AMAX is neither too
53: * large nor too small, it is not worth scaling by S.
54: *
55: * AMAX (output) DOUBLE PRECISION
56: * Absolute value of largest matrix element. If AMAX is very
57: * close to overflow or very close to underflow, the matrix
58: * should be scaled.
59: *
60: * INFO (output) INTEGER
61: * = 0: successful exit
62: * < 0: if INFO = -i, the i-th argument had an illegal value
63: * > 0: if INFO = i, the i-th diagonal element is nonpositive.
64: *
65: * =====================================================================
66: *
67: * .. Parameters ..
68: DOUBLE PRECISION ONE, ZERO
69: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
70: * ..
71: * .. Local Scalars ..
72: LOGICAL UPPER
73: INTEGER I, JJ
74: DOUBLE PRECISION SMIN
75: * ..
76: * .. External Functions ..
77: LOGICAL LSAME
78: EXTERNAL LSAME
79: * ..
80: * .. External Subroutines ..
81: EXTERNAL XERBLA
82: * ..
83: * .. Intrinsic Functions ..
84: INTRINSIC DBLE, MAX, MIN, SQRT
85: * ..
86: * .. Executable Statements ..
87: *
88: * Test the input parameters.
89: *
90: INFO = 0
91: UPPER = LSAME( UPLO, 'U' )
92: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
93: INFO = -1
94: ELSE IF( N.LT.0 ) THEN
95: INFO = -2
96: END IF
97: IF( INFO.NE.0 ) THEN
98: CALL XERBLA( 'ZPPEQU', -INFO )
99: RETURN
100: END IF
101: *
102: * Quick return if possible
103: *
104: IF( N.EQ.0 ) THEN
105: SCOND = ONE
106: AMAX = ZERO
107: RETURN
108: END IF
109: *
110: * Initialize SMIN and AMAX.
111: *
112: S( 1 ) = DBLE( AP( 1 ) )
113: SMIN = S( 1 )
114: AMAX = S( 1 )
115: *
116: IF( UPPER ) THEN
117: *
118: * UPLO = 'U': Upper triangle of A is stored.
119: * Find the minimum and maximum diagonal elements.
120: *
121: JJ = 1
122: DO 10 I = 2, N
123: JJ = JJ + I
124: S( I ) = DBLE( AP( JJ ) )
125: SMIN = MIN( SMIN, S( I ) )
126: AMAX = MAX( AMAX, S( I ) )
127: 10 CONTINUE
128: *
129: ELSE
130: *
131: * UPLO = 'L': Lower triangle of A is stored.
132: * Find the minimum and maximum diagonal elements.
133: *
134: JJ = 1
135: DO 20 I = 2, N
136: JJ = JJ + N - I + 2
137: S( I ) = DBLE( AP( JJ ) )
138: SMIN = MIN( SMIN, S( I ) )
139: AMAX = MAX( AMAX, S( I ) )
140: 20 CONTINUE
141: END IF
142: *
143: IF( SMIN.LE.ZERO ) THEN
144: *
145: * Find the first non-positive diagonal element and return.
146: *
147: DO 30 I = 1, N
148: IF( S( I ).LE.ZERO ) THEN
149: INFO = I
150: RETURN
151: END IF
152: 30 CONTINUE
153: ELSE
154: *
155: * Set the scale factors to the reciprocals
156: * of the diagonal elements.
157: *
158: DO 40 I = 1, N
159: S( I ) = ONE / SQRT( S( I ) )
160: 40 CONTINUE
161: *
162: * Compute SCOND = min(S(I)) / max(S(I))
163: *
164: SCOND = SQRT( SMIN ) / SQRT( AMAX )
165: END IF
166: RETURN
167: *
168: * End of ZPPEQU
169: *
170: END
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