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