Annotation of rpl/lapack/lapack/zhetd2.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZHETD2
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZHETD2 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetd2.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetd2.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetd2.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZHETD2( 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 D( * ), E( * )
! 29: * COMPLEX*16 A( LDA, * ), TAU( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> ZHETD2 reduces a complex Hermitian matrix A to real symmetric
! 39: *> tridiagonal form T by a unitary similarity transformation:
! 40: *> Q**H * A * Q = T.
! 41: *> \endverbatim
! 42: *
! 43: * Arguments:
! 44: * ==========
! 45: *
! 46: *> \param[in] UPLO
! 47: *> \verbatim
! 48: *> UPLO is CHARACTER*1
! 49: *> Specifies whether the upper or lower triangular part of the
! 50: *> Hermitian matrix A is stored:
! 51: *> = 'U': Upper triangular
! 52: *> = 'L': Lower triangular
! 53: *> \endverbatim
! 54: *>
! 55: *> \param[in] N
! 56: *> \verbatim
! 57: *> N is INTEGER
! 58: *> The order of the matrix A. N >= 0.
! 59: *> \endverbatim
! 60: *>
! 61: *> \param[in,out] A
! 62: *> \verbatim
! 63: *> A is COMPLEX*16 array, dimension (LDA,N)
! 64: *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 65: *> n-by-n upper triangular part of A contains the upper
! 66: *> triangular part of the matrix A, and the strictly lower
! 67: *> triangular part of A is not referenced. If UPLO = 'L', the
! 68: *> leading n-by-n lower triangular part of A contains the lower
! 69: *> triangular part of the matrix A, and the strictly upper
! 70: *> triangular part of A is not referenced.
! 71: *> On exit, if UPLO = 'U', the diagonal and first superdiagonal
! 72: *> of A are overwritten by the corresponding elements of the
! 73: *> tridiagonal matrix T, and the elements above the first
! 74: *> superdiagonal, with the array TAU, represent the unitary
! 75: *> matrix Q as a product of elementary reflectors; if UPLO
! 76: *> = 'L', the diagonal and first subdiagonal of A are over-
! 77: *> written by the corresponding elements of the tridiagonal
! 78: *> matrix T, and the elements below the first subdiagonal, with
! 79: *> the array TAU, represent the unitary matrix Q as a product
! 80: *> of elementary reflectors. See Further Details.
! 81: *> \endverbatim
! 82: *>
! 83: *> \param[in] LDA
! 84: *> \verbatim
! 85: *> LDA is INTEGER
! 86: *> The leading dimension of the array A. LDA >= max(1,N).
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[out] D
! 90: *> \verbatim
! 91: *> D is DOUBLE PRECISION array, dimension (N)
! 92: *> The diagonal elements of the tridiagonal matrix T:
! 93: *> D(i) = A(i,i).
! 94: *> \endverbatim
! 95: *>
! 96: *> \param[out] E
! 97: *> \verbatim
! 98: *> E is DOUBLE PRECISION array, dimension (N-1)
! 99: *> The off-diagonal elements of the tridiagonal matrix T:
! 100: *> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
! 101: *> \endverbatim
! 102: *>
! 103: *> \param[out] TAU
! 104: *> \verbatim
! 105: *> TAU is COMPLEX*16 array, dimension (N-1)
! 106: *> The scalar factors of the elementary reflectors (see Further
! 107: *> Details).
! 108: *> \endverbatim
! 109: *>
! 110: *> \param[out] INFO
! 111: *> \verbatim
! 112: *> INFO is INTEGER
! 113: *> = 0: successful exit
! 114: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 115: *> \endverbatim
! 116: *
! 117: * Authors:
! 118: * ========
! 119: *
! 120: *> \author Univ. of Tennessee
! 121: *> \author Univ. of California Berkeley
! 122: *> \author Univ. of Colorado Denver
! 123: *> \author NAG Ltd.
! 124: *
! 125: *> \date November 2011
! 126: *
! 127: *> \ingroup complex16HEcomputational
! 128: *
! 129: *> \par Further Details:
! 130: * =====================
! 131: *>
! 132: *> \verbatim
! 133: *>
! 134: *> If UPLO = 'U', the matrix Q is represented as a product of elementary
! 135: *> reflectors
! 136: *>
! 137: *> Q = H(n-1) . . . H(2) H(1).
! 138: *>
! 139: *> Each H(i) has the form
! 140: *>
! 141: *> H(i) = I - tau * v * v**H
! 142: *>
! 143: *> where tau is a complex scalar, and v is a complex vector with
! 144: *> v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
! 145: *> A(1:i-1,i+1), and tau in TAU(i).
! 146: *>
! 147: *> If UPLO = 'L', the matrix Q is represented as a product of elementary
! 148: *> reflectors
! 149: *>
! 150: *> Q = H(1) H(2) . . . H(n-1).
! 151: *>
! 152: *> Each H(i) has the form
! 153: *>
! 154: *> H(i) = I - tau * v * v**H
! 155: *>
! 156: *> where tau is a complex scalar, and v is a complex vector with
! 157: *> v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
! 158: *> and tau in TAU(i).
! 159: *>
! 160: *> The contents of A on exit are illustrated by the following examples
! 161: *> with n = 5:
! 162: *>
! 163: *> if UPLO = 'U': if UPLO = 'L':
! 164: *>
! 165: *> ( d e v2 v3 v4 ) ( d )
! 166: *> ( d e v3 v4 ) ( e d )
! 167: *> ( d e v4 ) ( v1 e d )
! 168: *> ( d e ) ( v1 v2 e d )
! 169: *> ( d ) ( v1 v2 v3 e d )
! 170: *>
! 171: *> where d and e denote diagonal and off-diagonal elements of T, and vi
! 172: *> denotes an element of the vector defining H(i).
! 173: *> \endverbatim
! 174: *>
! 175: * =====================================================================
1.1 bertrand 176: SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
177: *
1.9 ! bertrand 178: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 179: * -- LAPACK is a software package provided by Univ. of Tennessee, --
180: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 181: * November 2011
1.1 bertrand 182: *
183: * .. Scalar Arguments ..
184: CHARACTER UPLO
185: INTEGER INFO, LDA, N
186: * ..
187: * .. Array Arguments ..
188: DOUBLE PRECISION D( * ), E( * )
189: COMPLEX*16 A( LDA, * ), TAU( * )
190: * ..
191: *
192: * =====================================================================
193: *
194: * .. Parameters ..
195: COMPLEX*16 ONE, ZERO, HALF
196: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
197: $ ZERO = ( 0.0D+0, 0.0D+0 ),
198: $ HALF = ( 0.5D+0, 0.0D+0 ) )
199: * ..
200: * .. Local Scalars ..
201: LOGICAL UPPER
202: INTEGER I
203: COMPLEX*16 ALPHA, TAUI
204: * ..
205: * .. External Subroutines ..
206: EXTERNAL XERBLA, ZAXPY, ZHEMV, ZHER2, ZLARFG
207: * ..
208: * .. External Functions ..
209: LOGICAL LSAME
210: COMPLEX*16 ZDOTC
211: EXTERNAL LSAME, ZDOTC
212: * ..
213: * .. Intrinsic Functions ..
214: INTRINSIC DBLE, MAX, MIN
215: * ..
216: * .. Executable Statements ..
217: *
218: * Test the input parameters
219: *
220: INFO = 0
1.8 bertrand 221: UPPER = LSAME( UPLO, 'U')
1.1 bertrand 222: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
223: INFO = -1
224: ELSE IF( N.LT.0 ) THEN
225: INFO = -2
226: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
227: INFO = -4
228: END IF
229: IF( INFO.NE.0 ) THEN
230: CALL XERBLA( 'ZHETD2', -INFO )
231: RETURN
232: END IF
233: *
234: * Quick return if possible
235: *
236: IF( N.LE.0 )
237: $ RETURN
238: *
239: IF( UPPER ) THEN
240: *
241: * Reduce the upper triangle of A
242: *
243: A( N, N ) = DBLE( A( N, N ) )
244: DO 10 I = N - 1, 1, -1
245: *
1.8 bertrand 246: * Generate elementary reflector H(i) = I - tau * v * v**H
1.1 bertrand 247: * to annihilate A(1:i-1,i+1)
248: *
249: ALPHA = A( I, I+1 )
250: CALL ZLARFG( I, ALPHA, A( 1, I+1 ), 1, TAUI )
251: E( I ) = ALPHA
252: *
253: IF( TAUI.NE.ZERO ) THEN
254: *
255: * Apply H(i) from both sides to A(1:i,1:i)
256: *
257: A( I, I+1 ) = ONE
258: *
259: * Compute x := tau * A * v storing x in TAU(1:i)
260: *
261: CALL ZHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
262: $ TAU, 1 )
263: *
1.8 bertrand 264: * Compute w := x - 1/2 * tau * (x**H * v) * v
1.1 bertrand 265: *
266: ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, A( 1, I+1 ), 1 )
267: CALL ZAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
268: *
269: * Apply the transformation as a rank-2 update:
1.8 bertrand 270: * A := A - v * w**H - w * v**H
1.1 bertrand 271: *
272: CALL ZHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
273: $ LDA )
274: *
275: ELSE
276: A( I, I ) = DBLE( A( I, I ) )
277: END IF
278: A( I, I+1 ) = E( I )
279: D( I+1 ) = A( I+1, I+1 )
280: TAU( I ) = TAUI
281: 10 CONTINUE
282: D( 1 ) = A( 1, 1 )
283: ELSE
284: *
285: * Reduce the lower triangle of A
286: *
287: A( 1, 1 ) = DBLE( A( 1, 1 ) )
288: DO 20 I = 1, N - 1
289: *
1.8 bertrand 290: * Generate elementary reflector H(i) = I - tau * v * v**H
1.1 bertrand 291: * to annihilate A(i+2:n,i)
292: *
293: ALPHA = A( I+1, I )
294: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAUI )
295: E( I ) = ALPHA
296: *
297: IF( TAUI.NE.ZERO ) THEN
298: *
299: * Apply H(i) from both sides to A(i+1:n,i+1:n)
300: *
301: A( I+1, I ) = ONE
302: *
303: * Compute x := tau * A * v storing y in TAU(i:n-1)
304: *
305: CALL ZHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
306: $ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
307: *
1.8 bertrand 308: * Compute w := x - 1/2 * tau * (x**H * v) * v
1.1 bertrand 309: *
310: ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, A( I+1, I ),
311: $ 1 )
312: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
313: *
314: * Apply the transformation as a rank-2 update:
1.8 bertrand 315: * A := A - v * w**H - w * v**H
1.1 bertrand 316: *
317: CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
318: $ A( I+1, I+1 ), LDA )
319: *
320: ELSE
321: A( I+1, I+1 ) = DBLE( A( I+1, I+1 ) )
322: END IF
323: A( I+1, I ) = E( I )
324: D( I ) = A( I, I )
325: TAU( I ) = TAUI
326: 20 CONTINUE
327: D( N ) = A( N, N )
328: END IF
329: *
330: RETURN
331: *
332: * End of ZHETD2
333: *
334: END
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