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
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11: *> [TGZ]</a>
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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: *> \ingroup doubleOTHERcomputational
115: *
116: *> \par Further Details:
117: * =====================
118: *>
119: *> \verbatim
120: *>
121: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
122: *> reflectors
123: *>
124: *> Q = H(n-1) . . . H(2) H(1).
125: *>
126: *> Each H(i) has the form
127: *>
128: *> H(i) = I - tau * v * v**T
129: *>
130: *> where tau is a real scalar, and v is a real vector with
131: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
132: *> overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
133: *>
134: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
135: *> reflectors
136: *>
137: *> Q = H(1) H(2) . . . H(n-1).
138: *>
139: *> Each H(i) has the form
140: *>
141: *> H(i) = I - tau * v * v**T
142: *>
143: *> where tau is a real scalar, and v is a real vector with
144: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
145: *> overwriting A(i+2:n,i), and tau is stored in TAU(i).
146: *> \endverbatim
147: *>
148: * =====================================================================
149: SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
150: *
151: * -- LAPACK computational routine --
152: * -- LAPACK is a software package provided by Univ. of Tennessee, --
153: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
154: *
155: * .. Scalar Arguments ..
156: CHARACTER UPLO
157: INTEGER INFO, N
158: * ..
159: * .. Array Arguments ..
160: DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * )
161: * ..
162: *
163: * =====================================================================
164: *
165: * .. Parameters ..
166: DOUBLE PRECISION ONE, ZERO, HALF
167: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
168: $ HALF = 1.0D0 / 2.0D0 )
169: * ..
170: * .. Local Scalars ..
171: LOGICAL UPPER
172: INTEGER I, I1, I1I1, II
173: DOUBLE PRECISION ALPHA, TAUI
174: * ..
175: * .. External Subroutines ..
176: EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
177: * ..
178: * .. External Functions ..
179: LOGICAL LSAME
180: DOUBLE PRECISION DDOT
181: EXTERNAL LSAME, DDOT
182: * ..
183: * .. Executable Statements ..
184: *
185: * Test the input parameters
186: *
187: INFO = 0
188: UPPER = LSAME( UPLO, 'U' )
189: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
190: INFO = -1
191: ELSE IF( N.LT.0 ) THEN
192: INFO = -2
193: END IF
194: IF( INFO.NE.0 ) THEN
195: CALL XERBLA( 'DSPTRD', -INFO )
196: RETURN
197: END IF
198: *
199: * Quick return if possible
200: *
201: IF( N.LE.0 )
202: $ RETURN
203: *
204: IF( UPPER ) THEN
205: *
206: * Reduce the upper triangle of A.
207: * I1 is the index in AP of A(1,I+1).
208: *
209: I1 = N*( N-1 ) / 2 + 1
210: DO 10 I = N - 1, 1, -1
211: *
212: * Generate elementary reflector H(i) = I - tau * v * v**T
213: * to annihilate A(1:i-1,i+1)
214: *
215: CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
216: E( I ) = AP( I1+I-1 )
217: *
218: IF( TAUI.NE.ZERO ) THEN
219: *
220: * Apply H(i) from both sides to A(1:i,1:i)
221: *
222: AP( I1+I-1 ) = ONE
223: *
224: * Compute y := tau * A * v storing y in TAU(1:i)
225: *
226: CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
227: $ 1 )
228: *
229: * Compute w := y - 1/2 * tau * (y**T *v) * v
230: *
231: ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
232: CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
233: *
234: * Apply the transformation as a rank-2 update:
235: * A := A - v * w**T - w * v**T
236: *
237: CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
238: *
239: AP( I1+I-1 ) = E( I )
240: END IF
241: D( I+1 ) = AP( I1+I )
242: TAU( I ) = TAUI
243: I1 = I1 - I
244: 10 CONTINUE
245: D( 1 ) = AP( 1 )
246: ELSE
247: *
248: * Reduce the lower triangle of A. II is the index in AP of
249: * A(i,i) and I1I1 is the index of A(i+1,i+1).
250: *
251: II = 1
252: DO 20 I = 1, N - 1
253: I1I1 = II + N - I + 1
254: *
255: * Generate elementary reflector H(i) = I - tau * v * v**T
256: * to annihilate A(i+2:n,i)
257: *
258: CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
259: E( I ) = AP( II+1 )
260: *
261: IF( TAUI.NE.ZERO ) THEN
262: *
263: * Apply H(i) from both sides to A(i+1:n,i+1:n)
264: *
265: AP( II+1 ) = ONE
266: *
267: * Compute y := tau * A * v storing y in TAU(i:n-1)
268: *
269: CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
270: $ ZERO, TAU( I ), 1 )
271: *
272: * Compute w := y - 1/2 * tau * (y**T *v) * v
273: *
274: ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
275: $ 1 )
276: CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
277: *
278: * Apply the transformation as a rank-2 update:
279: * A := A - v * w**T - w * v**T
280: *
281: CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
282: $ AP( I1I1 ) )
283: *
284: AP( II+1 ) = E( I )
285: END IF
286: D( I ) = AP( II )
287: TAU( I ) = TAUI
288: II = I1I1
289: 20 CONTINUE
290: D( N ) = AP( II )
291: END IF
292: *
293: RETURN
294: *
295: * End of DSPTRD
296: *
297: END
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