Annotation of rpl/lapack/lapack/dsptrd.f, revision 1.11
1.9 bertrand 1: *> \brief \b DSPTRD
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSPTRD + dependencies
10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.f">
11: *> [TGZ]</a>
12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsptrd.f">
13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsptrd.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DSPTRD reduces a real symmetric matrix A stored in packed form to
38: *> symmetric tridiagonal form T by an orthogonal similarity
39: *> transformation: Q**T * A * Q = T.
40: *> \endverbatim
41: *
42: * Arguments:
43: * ==========
44: *
45: *> \param[in] UPLO
46: *> \verbatim
47: *> UPLO is CHARACTER*1
48: *> = 'U': Upper triangle of A is stored;
49: *> = 'L': Lower triangle of A is stored.
50: *> \endverbatim
51: *>
52: *> \param[in] N
53: *> \verbatim
54: *> N is INTEGER
55: *> The order of the matrix A. N >= 0.
56: *> \endverbatim
57: *>
58: *> \param[in,out] AP
59: *> \verbatim
60: *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
61: *> On entry, the upper or lower triangle of the symmetric matrix
62: *> A, packed columnwise in a linear array. The j-th column of A
63: *> is stored in the array AP as follows:
64: *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
65: *> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
66: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
67: *> of A are overwritten by the corresponding elements of the
68: *> tridiagonal matrix T, and the elements above the first
69: *> superdiagonal, with the array TAU, represent the orthogonal
70: *> matrix Q as a product of elementary reflectors; if UPLO
71: *> = 'L', the diagonal and first subdiagonal of A are over-
72: *> written by the corresponding elements of the tridiagonal
73: *> matrix T, and the elements below the first subdiagonal, with
74: *> the array TAU, represent the orthogonal matrix Q as a product
75: *> of elementary reflectors. See Further Details.
76: *> \endverbatim
77: *>
78: *> \param[out] D
79: *> \verbatim
80: *> D is DOUBLE PRECISION array, dimension (N)
81: *> The diagonal elements of the tridiagonal matrix T:
82: *> D(i) = A(i,i).
83: *> \endverbatim
84: *>
85: *> \param[out] E
86: *> \verbatim
87: *> E is DOUBLE PRECISION array, dimension (N-1)
88: *> The off-diagonal elements of the tridiagonal matrix T:
89: *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
90: *> \endverbatim
91: *>
92: *> \param[out] TAU
93: *> \verbatim
94: *> TAU is DOUBLE PRECISION array, dimension (N-1)
95: *> The scalar factors of the elementary reflectors (see Further
96: *> Details).
97: *> \endverbatim
98: *>
99: *> \param[out] INFO
100: *> \verbatim
101: *> INFO is INTEGER
102: *> = 0: successful exit
103: *> < 0: if INFO = -i, the i-th argument had an illegal value
104: *> \endverbatim
105: *
106: * Authors:
107: * ========
108: *
109: *> \author Univ. of Tennessee
110: *> \author Univ. of California Berkeley
111: *> \author Univ. of Colorado Denver
112: *> \author NAG Ltd.
113: *
114: *> \date November 2011
115: *
116: *> \ingroup doubleOTHERcomputational
117: *
118: *> \par Further Details:
119: * =====================
120: *>
121: *> \verbatim
122: *>
123: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
124: *> reflectors
125: *>
126: *> Q = H(n-1) . . . H(2) H(1).
127: *>
128: *> Each H(i) has the form
129: *>
130: *> H(i) = I - tau * v * v**T
131: *>
132: *> where tau is a real scalar, and v is a real vector with
133: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
134: *> overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
135: *>
136: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
137: *> reflectors
138: *>
139: *> Q = H(1) H(2) . . . H(n-1).
140: *>
141: *> Each H(i) has the form
142: *>
143: *> H(i) = I - tau * v * v**T
144: *>
145: *> where tau is a real scalar, and v is a real vector with
146: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
147: *> overwriting A(i+2:n,i), and tau is stored in TAU(i).
148: *> \endverbatim
149: *>
150: * =====================================================================
1.1 bertrand 151: SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
152: *
1.9 bertrand 153: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 154: * -- LAPACK is a software package provided by Univ. of Tennessee, --
155: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 bertrand 156: * November 2011
1.1 bertrand 157: *
158: * .. Scalar Arguments ..
159: CHARACTER UPLO
160: INTEGER INFO, N
161: * ..
162: * .. Array Arguments ..
163: DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
164: * ..
165: *
166: * =====================================================================
167: *
168: * .. Parameters ..
169: DOUBLE PRECISION ONE, ZERO, HALF
170: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
171: $ HALF = 1.0D0 / 2.0D0 )
172: * ..
173: * .. Local Scalars ..
174: LOGICAL UPPER
175: INTEGER I, I1, I1I1, II
176: DOUBLE PRECISION ALPHA, TAUI
177: * ..
178: * .. External Subroutines ..
179: EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
180: * ..
181: * .. External Functions ..
182: LOGICAL LSAME
183: DOUBLE PRECISION DDOT
184: EXTERNAL LSAME, DDOT
185: * ..
186: * .. Executable Statements ..
187: *
188: * Test the input parameters
189: *
190: INFO = 0
191: UPPER = LSAME( UPLO, 'U' )
192: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
193: INFO = -1
194: ELSE IF( N.LT.0 ) THEN
195: INFO = -2
196: END IF
197: IF( INFO.NE.0 ) THEN
198: CALL XERBLA( 'DSPTRD', -INFO )
199: RETURN
200: END IF
201: *
202: * Quick return if possible
203: *
204: IF( N.LE.0 )
205: $ RETURN
206: *
207: IF( UPPER ) THEN
208: *
209: * Reduce the upper triangle of A.
210: * I1 is the index in AP of A(1,I+1).
211: *
212: I1 = N*( N-1 ) / 2 + 1
213: DO 10 I = N - 1, 1, -1
214: *
1.8 bertrand 215: * Generate elementary reflector H(i) = I - tau * v * v**T
1.1 bertrand 216: * to annihilate A(1:i-1,i+1)
217: *
218: CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
219: E( I ) = AP( I1+I-1 )
220: *
221: IF( TAUI.NE.ZERO ) THEN
222: *
223: * Apply H(i) from both sides to A(1:i,1:i)
224: *
225: AP( I1+I-1 ) = ONE
226: *
227: * Compute y := tau * A * v storing y in TAU(1:i)
228: *
229: CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
230: $ 1 )
231: *
1.8 bertrand 232: * Compute w := y - 1/2 * tau * (y**T *v) * v
1.1 bertrand 233: *
234: ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
235: CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
236: *
237: * Apply the transformation as a rank-2 update:
1.8 bertrand 238: * A := A - v * w**T - w * v**T
1.1 bertrand 239: *
240: CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
241: *
242: AP( I1+I-1 ) = E( I )
243: END IF
244: D( I+1 ) = AP( I1+I )
245: TAU( I ) = TAUI
246: I1 = I1 - I
247: 10 CONTINUE
248: D( 1 ) = AP( 1 )
249: ELSE
250: *
251: * Reduce the lower triangle of A. II is the index in AP of
252: * A(i,i) and I1I1 is the index of A(i+1,i+1).
253: *
254: II = 1
255: DO 20 I = 1, N - 1
256: I1I1 = II + N - I + 1
257: *
1.8 bertrand 258: * Generate elementary reflector H(i) = I - tau * v * v**T
1.1 bertrand 259: * to annihilate A(i+2:n,i)
260: *
261: CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
262: E( I ) = AP( II+1 )
263: *
264: IF( TAUI.NE.ZERO ) THEN
265: *
266: * Apply H(i) from both sides to A(i+1:n,i+1:n)
267: *
268: AP( II+1 ) = ONE
269: *
270: * Compute y := tau * A * v storing y in TAU(i:n-1)
271: *
272: CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
273: $ ZERO, TAU( I ), 1 )
274: *
1.8 bertrand 275: * Compute w := y - 1/2 * tau * (y**T *v) * v
1.1 bertrand 276: *
277: ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
278: $ 1 )
279: CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
280: *
281: * Apply the transformation as a rank-2 update:
1.8 bertrand 282: * A := A - v * w**T - w * v**T
1.1 bertrand 283: *
284: CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
285: $ AP( I1I1 ) )
286: *
287: AP( II+1 ) = E( I )
288: END IF
289: D( I ) = AP( II )
290: TAU( I ) = TAUI
291: II = I1I1
292: 20 CONTINUE
293: D( N ) = AP( II )
294: END IF
295: *
296: RETURN
297: *
298: * End of DSPTRD
299: *
300: END
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