1: DOUBLE PRECISION FUNCTION ZLANHB( 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: * ZLANHB 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 hermitian band matrix A, with k super-diagonals.
24: *
25: * Description
26: * ===========
27: *
28: * ZLANHB returns the value
29: *
30: * ZLANHB = ( 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 ZLANHB 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
54: * = 'L': Lower triangular
55: *
56: * N (input) INTEGER
57: * The order of the matrix A. N >= 0. When N = 0, ZLANHB 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 hermitian 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: * Note that the imaginary parts of the diagonal elements need
71: * not be set and are assumed to be zero.
72: *
73: * LDAB (input) INTEGER
74: * The leading dimension of the array AB. LDAB >= K+1.
75: *
76: * WORK (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
77: * where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
78: * WORK is not referenced.
79: *
80: * =====================================================================
81: *
82: * .. Parameters ..
83: DOUBLE PRECISION ONE, ZERO
84: PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
85: * ..
86: * .. Local Scalars ..
87: INTEGER I, J, L
88: DOUBLE PRECISION ABSA, SCALE, SUM, VALUE
89: * ..
90: * .. External Functions ..
91: LOGICAL LSAME
92: EXTERNAL LSAME
93: * ..
94: * .. External Subroutines ..
95: EXTERNAL ZLASSQ
96: * ..
97: * .. Intrinsic Functions ..
98: INTRINSIC ABS, DBLE, MAX, MIN, SQRT
99: * ..
100: * .. Executable Statements ..
101: *
102: IF( N.EQ.0 ) THEN
103: VALUE = ZERO
104: ELSE IF( LSAME( NORM, 'M' ) ) THEN
105: *
106: * Find max(abs(A(i,j))).
107: *
108: VALUE = ZERO
109: IF( LSAME( UPLO, 'U' ) ) THEN
110: DO 20 J = 1, N
111: DO 10 I = MAX( K+2-J, 1 ), K
112: VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
113: 10 CONTINUE
114: VALUE = MAX( VALUE, ABS( DBLE( AB( K+1, J ) ) ) )
115: 20 CONTINUE
116: ELSE
117: DO 40 J = 1, N
118: VALUE = MAX( VALUE, ABS( DBLE( AB( 1, J ) ) ) )
119: DO 30 I = 2, MIN( N+1-J, K+1 )
120: VALUE = MAX( VALUE, ABS( AB( I, J ) ) )
121: 30 CONTINUE
122: 40 CONTINUE
123: END IF
124: ELSE IF( ( LSAME( NORM, 'I' ) ) .OR. ( LSAME( NORM, 'O' ) ) .OR.
125: $ ( NORM.EQ.'1' ) ) THEN
126: *
127: * Find normI(A) ( = norm1(A), since A is hermitian).
128: *
129: VALUE = ZERO
130: IF( LSAME( UPLO, 'U' ) ) THEN
131: DO 60 J = 1, N
132: SUM = ZERO
133: L = K + 1 - J
134: DO 50 I = MAX( 1, J-K ), J - 1
135: ABSA = ABS( AB( L+I, J ) )
136: SUM = SUM + ABSA
137: WORK( I ) = WORK( I ) + ABSA
138: 50 CONTINUE
139: WORK( J ) = SUM + ABS( DBLE( AB( K+1, J ) ) )
140: 60 CONTINUE
141: DO 70 I = 1, N
142: VALUE = MAX( VALUE, WORK( I ) )
143: 70 CONTINUE
144: ELSE
145: DO 80 I = 1, N
146: WORK( I ) = ZERO
147: 80 CONTINUE
148: DO 100 J = 1, N
149: SUM = WORK( J ) + ABS( DBLE( AB( 1, J ) ) )
150: L = 1 - J
151: DO 90 I = J + 1, MIN( N, J+K )
152: ABSA = ABS( AB( L+I, J ) )
153: SUM = SUM + ABSA
154: WORK( I ) = WORK( I ) + ABSA
155: 90 CONTINUE
156: VALUE = MAX( VALUE, SUM )
157: 100 CONTINUE
158: END IF
159: ELSE IF( ( LSAME( NORM, 'F' ) ) .OR. ( LSAME( NORM, 'E' ) ) ) THEN
160: *
161: * Find normF(A).
162: *
163: SCALE = ZERO
164: SUM = ONE
165: IF( K.GT.0 ) THEN
166: IF( LSAME( UPLO, 'U' ) ) THEN
167: DO 110 J = 2, N
168: CALL ZLASSQ( MIN( J-1, K ), AB( MAX( K+2-J, 1 ), J ),
169: $ 1, SCALE, SUM )
170: 110 CONTINUE
171: L = K + 1
172: ELSE
173: DO 120 J = 1, N - 1
174: CALL ZLASSQ( MIN( N-J, K ), AB( 2, J ), 1, SCALE,
175: $ SUM )
176: 120 CONTINUE
177: L = 1
178: END IF
179: SUM = 2*SUM
180: ELSE
181: L = 1
182: END IF
183: DO 130 J = 1, N
184: IF( DBLE( AB( L, J ) ).NE.ZERO ) THEN
185: ABSA = ABS( DBLE( AB( L, J ) ) )
186: IF( SCALE.LT.ABSA ) THEN
187: SUM = ONE + SUM*( SCALE / ABSA )**2
188: SCALE = ABSA
189: ELSE
190: SUM = SUM + ( ABSA / SCALE )**2
191: END IF
192: END IF
193: 130 CONTINUE
194: VALUE = SCALE*SQRT( SUM )
195: END IF
196: *
197: ZLANHB = VALUE
198: RETURN
199: *
200: * End of ZLANHB
201: *
202: END
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