1: *> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm).
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download DSYTD2 + dependencies
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15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
22: *
23: * .. Scalar Arguments ..
24: * CHARACTER UPLO
25: * INTEGER INFO, LDA, N
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
29: * ..
30: *
31: *
32: *> \par Purpose:
33: * =============
34: *>
35: *> \verbatim
36: *>
37: *> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
38: *> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
39: *> \endverbatim
40: *
41: * Arguments:
42: * ==========
43: *
44: *> \param[in] UPLO
45: *> \verbatim
46: *> UPLO is CHARACTER*1
47: *> Specifies whether the upper or lower triangular part of the
48: *> symmetric matrix A is stored:
49: *> = 'U': Upper triangular
50: *> = 'L': Lower triangular
51: *> \endverbatim
52: *>
53: *> \param[in] N
54: *> \verbatim
55: *> N is INTEGER
56: *> The order of the matrix A. N >= 0.
57: *> \endverbatim
58: *>
59: *> \param[in,out] A
60: *> \verbatim
61: *> A is DOUBLE PRECISION array, dimension (LDA,N)
62: *> On entry, the symmetric matrix A. If UPLO = 'U', the leading
63: *> n-by-n upper triangular part of A contains the upper
64: *> triangular part of the matrix A, and the strictly lower
65: *> triangular part of A is not referenced. If UPLO = 'L', the
66: *> leading n-by-n lower triangular part of A contains the lower
67: *> triangular part of the matrix A, and the strictly upper
68: *> triangular part of A is not referenced.
69: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
70: *> of A are overwritten by the corresponding elements of the
71: *> tridiagonal matrix T, and the elements above the first
72: *> superdiagonal, with the array TAU, represent the orthogonal
73: *> matrix Q as a product of elementary reflectors; if UPLO
74: *> = 'L', the diagonal and first subdiagonal of A are over-
75: *> written by the corresponding elements of the tridiagonal
76: *> matrix T, and the elements below the first subdiagonal, with
77: *> the array TAU, represent the orthogonal matrix Q as a product
78: *> of elementary reflectors. See Further Details.
79: *> \endverbatim
80: *>
81: *> \param[in] LDA
82: *> \verbatim
83: *> LDA is INTEGER
84: *> The leading dimension of the array A. LDA >= max(1,N).
85: *> \endverbatim
86: *>
87: *> \param[out] D
88: *> \verbatim
89: *> D is DOUBLE PRECISION array, dimension (N)
90: *> The diagonal elements of the tridiagonal matrix T:
91: *> D(i) = A(i,i).
92: *> \endverbatim
93: *>
94: *> \param[out] E
95: *> \verbatim
96: *> E is DOUBLE PRECISION array, dimension (N-1)
97: *> The off-diagonal elements of the tridiagonal matrix T:
98: *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
99: *> \endverbatim
100: *>
101: *> \param[out] TAU
102: *> \verbatim
103: *> TAU is DOUBLE PRECISION array, dimension (N-1)
104: *> The scalar factors of the elementary reflectors (see Further
105: *> Details).
106: *> \endverbatim
107: *>
108: *> \param[out] INFO
109: *> \verbatim
110: *> INFO is INTEGER
111: *> = 0: successful exit
112: *> < 0: if INFO = -i, the i-th argument had an illegal value.
113: *> \endverbatim
114: *
115: * Authors:
116: * ========
117: *
118: *> \author Univ. of Tennessee
119: *> \author Univ. of California Berkeley
120: *> \author Univ. of Colorado Denver
121: *> \author NAG Ltd.
122: *
123: *> \ingroup doubleSYcomputational
124: *
125: *> \par Further Details:
126: * =====================
127: *>
128: *> \verbatim
129: *>
130: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
131: *> reflectors
132: *>
133: *> Q = H(n-1) . . . H(2) H(1).
134: *>
135: *> Each H(i) has the form
136: *>
137: *> H(i) = I - tau * v * v**T
138: *>
139: *> where tau is a real scalar, and v is a real vector with
140: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
141: *> A(1:i-1,i+1), and tau in TAU(i).
142: *>
143: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
144: *> reflectors
145: *>
146: *> Q = H(1) H(2) . . . H(n-1).
147: *>
148: *> Each H(i) has the form
149: *>
150: *> H(i) = I - tau * v * v**T
151: *>
152: *> where tau is a real scalar, and v is a real vector with
153: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
154: *> and tau in TAU(i).
155: *>
156: *> The contents of A on exit are illustrated by the following examples
157: *> with n = 5:
158: *>
159: *> if UPLO = 'U': if UPLO = 'L':
160: *>
161: *> ( d e v2 v3 v4 ) ( d )
162: *> ( d e v3 v4 ) ( e d )
163: *> ( d e v4 ) ( v1 e d )
164: *> ( d e ) ( v1 v2 e d )
165: *> ( d ) ( v1 v2 v3 e d )
166: *>
167: *> where d and e denote diagonal and off-diagonal elements of T, and vi
168: *> denotes an element of the vector defining H(i).
169: *> \endverbatim
170: *>
171: * =====================================================================
172: SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
173: *
174: * -- LAPACK computational routine --
175: * -- LAPACK is a software package provided by Univ. of Tennessee, --
176: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177: *
178: * .. Scalar Arguments ..
179: CHARACTER UPLO
180: INTEGER INFO, LDA, N
181: * ..
182: * .. Array Arguments ..
183: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * )
184: * ..
185: *
186: * =====================================================================
187: *
188: * .. Parameters ..
189: DOUBLE PRECISION ONE, ZERO, HALF
190: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
191: $ HALF = 1.0D0 / 2.0D0 )
192: * ..
193: * .. Local Scalars ..
194: LOGICAL UPPER
195: INTEGER I
196: DOUBLE PRECISION ALPHA, TAUI
197: * ..
198: * .. External Subroutines ..
199: EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
200: * ..
201: * .. External Functions ..
202: LOGICAL LSAME
203: DOUBLE PRECISION DDOT
204: EXTERNAL LSAME, DDOT
205: * ..
206: * .. Intrinsic Functions ..
207: INTRINSIC MAX, MIN
208: * ..
209: * .. Executable Statements ..
210: *
211: * Test the input parameters
212: *
213: INFO = 0
214: UPPER = LSAME( UPLO, 'U' )
215: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
216: INFO = -1
217: ELSE IF( N.LT.0 ) THEN
218: INFO = -2
219: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
220: INFO = -4
221: END IF
222: IF( INFO.NE.0 ) THEN
223: CALL XERBLA( 'DSYTD2', -INFO )
224: RETURN
225: END IF
226: *
227: * Quick return if possible
228: *
229: IF( N.LE.0 )
230: $ RETURN
231: *
232: IF( UPPER ) THEN
233: *
234: * Reduce the upper triangle of A
235: *
236: DO 10 I = N - 1, 1, -1
237: *
238: * Generate elementary reflector H(i) = I - tau * v * v**T
239: * to annihilate A(1:i-1,i+1)
240: *
241: CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
242: E( I ) = A( I, I+1 )
243: *
244: IF( TAUI.NE.ZERO ) THEN
245: *
246: * Apply H(i) from both sides to A(1:i,1:i)
247: *
248: A( I, I+1 ) = ONE
249: *
250: * Compute x := tau * A * v storing x in TAU(1:i)
251: *
252: CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
253: $ TAU, 1 )
254: *
255: * Compute w := x - 1/2 * tau * (x**T * v) * v
256: *
257: ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
258: CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
259: *
260: * Apply the transformation as a rank-2 update:
261: * A := A - v * w**T - w * v**T
262: *
263: CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
264: $ LDA )
265: *
266: A( I, I+1 ) = E( I )
267: END IF
268: D( I+1 ) = A( I+1, I+1 )
269: TAU( I ) = TAUI
270: 10 CONTINUE
271: D( 1 ) = A( 1, 1 )
272: ELSE
273: *
274: * Reduce the lower triangle of A
275: *
276: DO 20 I = 1, N - 1
277: *
278: * Generate elementary reflector H(i) = I - tau * v * v**T
279: * to annihilate A(i+2:n,i)
280: *
281: CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
282: $ TAUI )
283: E( I ) = A( I+1, I )
284: *
285: IF( TAUI.NE.ZERO ) THEN
286: *
287: * Apply H(i) from both sides to A(i+1:n,i+1:n)
288: *
289: A( I+1, I ) = ONE
290: *
291: * Compute x := tau * A * v storing y in TAU(i:n-1)
292: *
293: CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
294: $ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
295: *
296: * Compute w := x - 1/2 * tau * (x**T * v) * v
297: *
298: ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
299: $ 1 )
300: CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
301: *
302: * Apply the transformation as a rank-2 update:
303: * A := A - v * w**T - w * v**T
304: *
305: CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
306: $ A( I+1, I+1 ), LDA )
307: *
308: A( I+1, I ) = E( I )
309: END IF
310: D( I ) = A( I, I )
311: TAU( I ) = TAUI
312: 20 CONTINUE
313: D( N ) = A( N, N )
314: END IF
315: *
316: RETURN
317: *
318: * End of DSYTD2
319: *
320: END
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