LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ chptrd()

 subroutine chptrd ( character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAU, integer INFO )

CHPTRD

Purpose:
``` CHPTRD reduces a complex Hermitian matrix A stored in packed form to
real symmetric tridiagonal form T by a unitary similarity
transformation: Q**H * A * Q = T.```
Parameters
 [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in,out] AP ``` AP is COMPLEX array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the Hermitian matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, if UPLO = 'U', the diagonal and first superdiagonal of A are overwritten by the corresponding elements of the tridiagonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors; if UPLO = 'L', the diagonal and first subdiagonal of A are over- written by the corresponding elements of the tridiagonal matrix T, and the elements below the first subdiagonal, with the array TAU, represent the unitary matrix Q as a product of elementary reflectors. See Further Details.``` [out] D ``` D is REAL array, dimension (N) The diagonal elements of the tridiagonal matrix T: D(i) = A(i,i).``` [out] E ``` E is REAL array, dimension (N-1) The off-diagonal elements of the tridiagonal matrix T: E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.``` [out] TAU ``` TAU is COMPLEX array, dimension (N-1) The scalar factors of the elementary reflectors (see Further Details).``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors

Q = H(n-1) . . . H(2) H(1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
overwriting A(1:i-1,i+1), and tau is stored in TAU(i).

If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors

Q = H(1) H(2) . . . H(n-1).

Each H(i) has the form

H(i) = I - tau * v * v**H

where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
overwriting A(i+2:n,i), and tau is stored in TAU(i).```

Definition at line 150 of file chptrd.f.

151*
152* -- LAPACK computational routine --
153* -- LAPACK is a software package provided by Univ. of Tennessee, --
154* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
155*
156* .. Scalar Arguments ..
157 CHARACTER UPLO
158 INTEGER INFO, N
159* ..
160* .. Array Arguments ..
161 REAL D( * ), E( * )
162 COMPLEX AP( * ), TAU( * )
163* ..
164*
165* =====================================================================
166*
167* .. Parameters ..
168 COMPLEX ONE, ZERO, HALF
169 parameter( one = ( 1.0e+0, 0.0e+0 ),
170 \$ zero = ( 0.0e+0, 0.0e+0 ),
171 \$ half = ( 0.5e+0, 0.0e+0 ) )
172* ..
173* .. Local Scalars ..
174 LOGICAL UPPER
175 INTEGER I, I1, I1I1, II
176 COMPLEX ALPHA, TAUI
177* ..
178* .. External Subroutines ..
179 EXTERNAL caxpy, chpmv, chpr2, clarfg, xerbla
180* ..
181* .. External Functions ..
182 LOGICAL LSAME
183 COMPLEX CDOTC
184 EXTERNAL lsame, cdotc
185* ..
186* .. Intrinsic Functions ..
187 INTRINSIC real
188* ..
189* .. Executable Statements ..
190*
191* Test the input parameters
192*
193 info = 0
194 upper = lsame( uplo, 'U' )
195 IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
196 info = -1
197 ELSE IF( n.LT.0 ) THEN
198 info = -2
199 END IF
200 IF( info.NE.0 ) THEN
201 CALL xerbla( 'CHPTRD', -info )
202 RETURN
203 END IF
204*
205* Quick return if possible
206*
207 IF( n.LE.0 )
208 \$ RETURN
209*
210 IF( upper ) THEN
211*
212* Reduce the upper triangle of A.
213* I1 is the index in AP of A(1,I+1).
214*
215 i1 = n*( n-1 ) / 2 + 1
216 ap( i1+n-1 ) = real( ap( i1+n-1 ) )
217 DO 10 i = n - 1, 1, -1
218*
219* Generate elementary reflector H(i) = I - tau * v * v**H
220* to annihilate A(1:i-1,i+1)
221*
222 alpha = ap( i1+i-1 )
223 CALL clarfg( i, alpha, ap( i1 ), 1, taui )
224 e( i ) = real( alpha )
225*
226 IF( taui.NE.zero ) THEN
227*
228* Apply H(i) from both sides to A(1:i,1:i)
229*
230 ap( i1+i-1 ) = one
231*
232* Compute y := tau * A * v storing y in TAU(1:i)
233*
234 CALL chpmv( uplo, i, taui, ap, ap( i1 ), 1, zero, tau,
235 \$ 1 )
236*
237* Compute w := y - 1/2 * tau * (y**H *v) * v
238*
239 alpha = -half*taui*cdotc( i, tau, 1, ap( i1 ), 1 )
240 CALL caxpy( i, alpha, ap( i1 ), 1, tau, 1 )
241*
242* Apply the transformation as a rank-2 update:
243* A := A - v * w**H - w * v**H
244*
245 CALL chpr2( uplo, i, -one, ap( i1 ), 1, tau, 1, ap )
246*
247 END IF
248 ap( i1+i-1 ) = e( i )
249 d( i+1 ) = real( ap( i1+i ) )
250 tau( i ) = taui
251 i1 = i1 - i
252 10 CONTINUE
253 d( 1 ) = real( ap( 1 ) )
254 ELSE
255*
256* Reduce the lower triangle of A. II is the index in AP of
257* A(i,i) and I1I1 is the index of A(i+1,i+1).
258*
259 ii = 1
260 ap( 1 ) = real( ap( 1 ) )
261 DO 20 i = 1, n - 1
262 i1i1 = ii + n - i + 1
263*
264* Generate elementary reflector H(i) = I - tau * v * v**H
265* to annihilate A(i+2:n,i)
266*
267 alpha = ap( ii+1 )
268 CALL clarfg( n-i, alpha, ap( ii+2 ), 1, taui )
269 e( i ) = real( alpha )
270*
271 IF( taui.NE.zero ) THEN
272*
273* Apply H(i) from both sides to A(i+1:n,i+1:n)
274*
275 ap( ii+1 ) = one
276*
277* Compute y := tau * A * v storing y in TAU(i:n-1)
278*
279 CALL chpmv( uplo, n-i, taui, ap( i1i1 ), ap( ii+1 ), 1,
280 \$ zero, tau( i ), 1 )
281*
282* Compute w := y - 1/2 * tau * (y**H *v) * v
283*
284 alpha = -half*taui*cdotc( n-i, tau( i ), 1, ap( ii+1 ),
285 \$ 1 )
286 CALL caxpy( n-i, alpha, ap( ii+1 ), 1, tau( i ), 1 )
287*
288* Apply the transformation as a rank-2 update:
289* A := A - v * w**H - w * v**H
290*
291 CALL chpr2( uplo, n-i, -one, ap( ii+1 ), 1, tau( i ), 1,
292 \$ ap( i1i1 ) )
293*
294 END IF
295 ap( ii+1 ) = e( i )
296 d( i ) = real( ap( ii ) )
297 tau( i ) = taui
298 ii = i1i1
299 20 CONTINUE
300 d( n ) = real( ap( ii ) )
301 END IF
302*
303 RETURN
304*
305* End of CHPTRD
306*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
complex function cdotc(N, CX, INCX, CY, INCY)
CDOTC
Definition: cdotc.f:83
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine chpr2(UPLO, N, ALPHA, X, INCX, Y, INCY, AP)
CHPR2
Definition: chpr2.f:145
subroutine chpmv(UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CHPMV
Definition: chpmv.f:149
subroutine clarfg(N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
Definition: clarfg.f:106
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