Annotation of rpl/lapack/lapack/dsptrd.f, revision 1.7
1.1 bertrand 1: SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, 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: * ..
12: * .. Array Arguments ..
13: DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
14: * ..
15: *
16: * Purpose
17: * =======
18: *
19: * DSPTRD reduces a real symmetric matrix A stored in packed form to
20: * symmetric tridiagonal form T by an orthogonal similarity
21: * transformation: Q**T * A * Q = T.
22: *
23: * Arguments
24: * =========
25: *
26: * UPLO (input) CHARACTER*1
27: * = 'U': Upper triangle of A is stored;
28: * = 'L': Lower triangle of A is stored.
29: *
30: * N (input) INTEGER
31: * The order of the matrix A. N >= 0.
32: *
33: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
34: * On entry, the upper or lower triangle of the symmetric matrix
35: * A, packed columnwise in a linear array. The j-th column of A
36: * is stored in the array AP as follows:
37: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
38: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
39: * On exit, if UPLO = 'U', the diagonal and first superdiagonal
40: * of A are overwritten by the corresponding elements of the
41: * tridiagonal matrix T, and the elements above the first
42: * superdiagonal, with the array TAU, represent the orthogonal
43: * matrix Q as a product of elementary reflectors; if UPLO
44: * = 'L', the diagonal and first subdiagonal of A are over-
45: * written by the corresponding elements of the tridiagonal
46: * matrix T, and the elements below the first subdiagonal, with
47: * the array TAU, represent the orthogonal matrix Q as a product
48: * of elementary reflectors. See Further Details.
49: *
50: * D (output) DOUBLE PRECISION array, dimension (N)
51: * The diagonal elements of the tridiagonal matrix T:
52: * D(i) = A(i,i).
53: *
54: * E (output) DOUBLE PRECISION array, dimension (N-1)
55: * The off-diagonal elements of the tridiagonal matrix T:
56: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
57: *
58: * TAU (output) DOUBLE PRECISION array, dimension (N-1)
59: * The scalar factors of the elementary reflectors (see Further
60: * Details).
61: *
62: * INFO (output) INTEGER
63: * = 0: successful exit
64: * < 0: if INFO = -i, the i-th argument had an illegal value
65: *
66: * Further Details
67: * ===============
68: *
69: * If UPLO = 'U', the matrix Q is represented as a product of elementary
70: * reflectors
71: *
72: * Q = H(n-1) . . . H(2) H(1).
73: *
74: * Each H(i) has the form
75: *
76: * H(i) = I - tau * v * v'
77: *
78: * where tau is a real scalar, and v is a real vector with
79: * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
80: * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
81: *
82: * If UPLO = 'L', the matrix Q is represented as a product of elementary
83: * reflectors
84: *
85: * Q = H(1) H(2) . . . H(n-1).
86: *
87: * Each H(i) has the form
88: *
89: * H(i) = I - tau * v * v'
90: *
91: * where tau is a real scalar, and v is a real vector with
92: * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
93: * overwriting A(i+2:n,i), and tau is stored in TAU(i).
94: *
95: * =====================================================================
96: *
97: * .. Parameters ..
98: DOUBLE PRECISION ONE, ZERO, HALF
99: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
100: $ HALF = 1.0D0 / 2.0D0 )
101: * ..
102: * .. Local Scalars ..
103: LOGICAL UPPER
104: INTEGER I, I1, I1I1, II
105: DOUBLE PRECISION ALPHA, TAUI
106: * ..
107: * .. External Subroutines ..
108: EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
109: * ..
110: * .. External Functions ..
111: LOGICAL LSAME
112: DOUBLE PRECISION DDOT
113: EXTERNAL LSAME, DDOT
114: * ..
115: * .. Executable Statements ..
116: *
117: * Test the input parameters
118: *
119: INFO = 0
120: UPPER = LSAME( UPLO, 'U' )
121: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
122: INFO = -1
123: ELSE IF( N.LT.0 ) THEN
124: INFO = -2
125: END IF
126: IF( INFO.NE.0 ) THEN
127: CALL XERBLA( 'DSPTRD', -INFO )
128: RETURN
129: END IF
130: *
131: * Quick return if possible
132: *
133: IF( N.LE.0 )
134: $ RETURN
135: *
136: IF( UPPER ) THEN
137: *
138: * Reduce the upper triangle of A.
139: * I1 is the index in AP of A(1,I+1).
140: *
141: I1 = N*( N-1 ) / 2 + 1
142: DO 10 I = N - 1, 1, -1
143: *
144: * Generate elementary reflector H(i) = I - tau * v * v'
145: * to annihilate A(1:i-1,i+1)
146: *
147: CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
148: E( I ) = AP( I1+I-1 )
149: *
150: IF( TAUI.NE.ZERO ) THEN
151: *
152: * Apply H(i) from both sides to A(1:i,1:i)
153: *
154: AP( I1+I-1 ) = ONE
155: *
156: * Compute y := tau * A * v storing y in TAU(1:i)
157: *
158: CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
159: $ 1 )
160: *
161: * Compute w := y - 1/2 * tau * (y'*v) * v
162: *
163: ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
164: CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
165: *
166: * Apply the transformation as a rank-2 update:
167: * A := A - v * w' - w * v'
168: *
169: CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
170: *
171: AP( I1+I-1 ) = E( I )
172: END IF
173: D( I+1 ) = AP( I1+I )
174: TAU( I ) = TAUI
175: I1 = I1 - I
176: 10 CONTINUE
177: D( 1 ) = AP( 1 )
178: ELSE
179: *
180: * Reduce the lower triangle of A. II is the index in AP of
181: * A(i,i) and I1I1 is the index of A(i+1,i+1).
182: *
183: II = 1
184: DO 20 I = 1, N - 1
185: I1I1 = II + N - I + 1
186: *
187: * Generate elementary reflector H(i) = I - tau * v * v'
188: * to annihilate A(i+2:n,i)
189: *
190: CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
191: E( I ) = AP( II+1 )
192: *
193: IF( TAUI.NE.ZERO ) THEN
194: *
195: * Apply H(i) from both sides to A(i+1:n,i+1:n)
196: *
197: AP( II+1 ) = ONE
198: *
199: * Compute y := tau * A * v storing y in TAU(i:n-1)
200: *
201: CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
202: $ ZERO, TAU( I ), 1 )
203: *
204: * Compute w := y - 1/2 * tau * (y'*v) * v
205: *
206: ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
207: $ 1 )
208: CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
209: *
210: * Apply the transformation as a rank-2 update:
211: * A := A - v * w' - w * v'
212: *
213: CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
214: $ AP( I1I1 ) )
215: *
216: AP( II+1 ) = E( I )
217: END IF
218: D( I ) = AP( II )
219: TAU( I ) = TAUI
220: II = I1I1
221: 20 CONTINUE
222: D( N ) = AP( II )
223: END IF
224: *
225: RETURN
226: *
227: * End of DSPTRD
228: *
229: END
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