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1.1 bertrand 1: DOUBLE PRECISION FUNCTION DLANSB( NORM, UPLO, N, K, AB, LDAB,
2: $ WORK )
3: *
4: * -- LAPACK auxiliary routine (version 3.2) --
5: * -- LAPACK is a software package provided by Univ. of Tennessee, --
6: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
7: * November 2006
8: *
9: * .. Scalar Arguments ..
10: CHARACTER NORM, UPLO
11: INTEGER K, LDAB, N
12: * ..
13: * .. Array Arguments ..
14: DOUBLE PRECISION AB( LDAB, * ), WORK( * )
15: * ..
16: *
17: * Purpose
18: * =======
19: *
20: * DLANSB returns the value of the one norm, or the Frobenius norm, or
21: * the infinity norm, or the element of largest absolute value of an
22: * n by n symmetric band matrix A, with k super-diagonals.
23: *
24: * Description
25: * ===========
26: *
27: * DLANSB returns the value
28: *
29: * DLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
30: * (
31: * ( norm1(A), NORM = '1', 'O' or 'o'
32: * (
33: * ( normI(A), NORM = 'I' or 'i'
34: * (
35: * ( normF(A), NORM = 'F', 'f', 'E' or 'e'
36: *
37: * where norm1 denotes the one norm of a matrix (maximum column sum),
38: * normI denotes the infinity norm of a matrix (maximum row sum) and
39: * normF denotes the Frobenius norm of a matrix (square root of sum of
40: * squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
41: *
42: * Arguments
43: * =========
44: *
45: * NORM (input) CHARACTER*1
46: * Specifies the value to be returned in DLANSB as described
47: * above.
48: *
49: * UPLO (input) CHARACTER*1
50: * Specifies whether the upper or lower triangular part of the
51: * band matrix A is supplied.
52: * = 'U': Upper triangular part is supplied
53: * = 'L': Lower triangular part is supplied
54: *
55: * N (input) INTEGER
56: * The order of the matrix A. N >= 0. When N = 0, DLANSB is
57: * set to zero.
58: *
59: * K (input) INTEGER
60: * The number of super-diagonals or sub-diagonals of the
61: * band matrix A. K >= 0.
62: *
63: * AB (input) DOUBLE PRECISION array, dimension (LDAB,N)
64: * The upper or lower triangle of the symmetric band matrix A,
65: * stored in the first K+1 rows of AB. The j-th column of A is
66: * stored in the j-th column of the array AB as follows:
67: * if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
68: * if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
69: *
70: * LDAB (input) INTEGER
71: * The leading dimension of the array AB. LDAB >= K+1.
72: *
73: * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
74: * where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
75: * WORK is not referenced.
76: *
77: * =====================================================================
78: *
79: * .. Parameters ..
80: DOUBLE PRECISION ONE, ZERO
81: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
82: * ..
83: * .. Local Scalars ..
84: INTEGER I, J, L
85: DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
86: * ..
87: * .. External Subroutines ..
88: EXTERNAL DLASSQ
89: * ..
90: * .. External Functions ..
91: LOGICAL LSAME
92: EXTERNAL LSAME
93: * ..
94: * .. Intrinsic Functions ..
95: INTRINSIC ABS, MAX, MIN, SQRT
96: * ..
97: * .. Executable Statements ..
98: *
99: IF( N.EQ.0 ) THEN
100: VALUE = ZERO
101: ELSE IF( LSAME( NORM, 'M' ) ) THEN
102: *
103: * Find max(abs(A(i,j))).
104: *
105: VALUE = ZERO
106: IF( LSAME( UPLO, 'U' ) ) THEN
107: DO 20 J = 1, N
108: DO 10 I = MAX( K+2-J, 1 ), K + 1
109: VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
110: 10 CONTINUE
111: 20 CONTINUE
112: ELSE
113: DO 40 J = 1, N
114: DO 30 I = 1, MIN( N+1-J, K+1 )
115: VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
116: 30 CONTINUE
117: 40 CONTINUE
118: END IF
119: ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
120: $ ( NORM.EQ.'1' ) ) THEN
121: *
122: * Find normI(A) ( = norm1(A), since A is symmetric).
123: *
124: VALUE = ZERO
125: IF( LSAME( UPLO, 'U' ) ) THEN
126: DO 60 J = 1, N
127: SUM = ZERO
128: L = K + 1 - J
129: DO 50 I = MAX( 1, J-K ), J - 1
130: ABSA = ABS( AB( L+I, J ) )
131: SUM = SUM + ABSA
132: WORK( I ) = WORK( I ) + ABSA
133: 50 CONTINUE
134: WORK( J ) = SUM + ABS( AB( K+1, J ) )
135: 60 CONTINUE
136: DO 70 I = 1, N
137: VALUE = MAX( VALUE, WORK( I ) )
138: 70 CONTINUE
139: ELSE
140: DO 80 I = 1, N
141: WORK( I ) = ZERO
142: 80 CONTINUE
143: DO 100 J = 1, N
144: SUM = WORK( J ) + ABS( AB( 1, J ) )
145: L = 1 - J
146: DO 90 I = J + 1, MIN( N, J+K )
147: ABSA = ABS( AB( L+I, J ) )
148: SUM = SUM + ABSA
149: WORK( I ) = WORK( I ) + ABSA
150: 90 CONTINUE
151: VALUE = MAX( VALUE, SUM )
152: 100 CONTINUE
153: END IF
154: ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
155: *
156: * Find normF(A).
157: *
158: SCALE = ZERO
159: SUM = ONE
160: IF( K.GT.0 ) THEN
161: IF( LSAME( UPLO, 'U' ) ) THEN
162: DO 110 J = 2, N
163: CALL DLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
164: $ 1, SCALE, SUM )
165: 110 CONTINUE
166: L = K + 1
167: ELSE
168: DO 120 J = 1, N - 1
169: CALL DLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
170: $ SUM )
171: 120 CONTINUE
172: L = 1
173: END IF
174: SUM = 2*SUM
175: ELSE
176: L = 1
177: END IF
178: CALL DLASSQ( N, AB( L, 1 ), LDAB, SCALE, SUM )
179: VALUE = SCALE*SQRT( SUM )
180: END IF
181: *
182: DLANSB = VALUE
183: RETURN
184: *
185: * End of DLANSB
186: *
187: END