 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sqrt02()

 subroutine sqrt02 ( integer M, integer N, integer K, real, dimension( lda, * ) A, real, dimension( lda, * ) AF, real, dimension( lda, * ) Q, real, dimension( lda, * ) R, integer LDA, real, dimension( * ) TAU, real, dimension( lwork ) WORK, integer LWORK, real, dimension( * ) RWORK, real, dimension( * ) RESULT )

SQRT02

Purpose:
``` SQRT02 tests SORGQR, which generates an m-by-n matrix Q with
orthonornmal columns that is defined as the product of k elementary
reflectors.

Given the QR factorization of an m-by-n matrix A, SQRT02 generates
the orthogonal matrix Q defined by the factorization of the first k
columns of A; it compares R(1:n,1:k) with Q(1:m,1:n)'*A(1:m,1:k),
and checks that the columns of Q are orthonormal.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix Q to be generated. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix Q to be generated. M >= N >= 0.``` [in] K ``` K is INTEGER The number of elementary reflectors whose product defines the matrix Q. N >= K >= 0.``` [in] A ``` A is REAL array, dimension (LDA,N) The m-by-n matrix A which was factorized by SQRT01.``` [in] AF ``` AF is REAL array, dimension (LDA,N) Details of the QR factorization of A, as returned by SGEQRF. See SGEQRF for further details.``` [out] Q ` Q is REAL array, dimension (LDA,N)` [out] R ` R is REAL array, dimension (LDA,N)` [in] LDA ``` LDA is INTEGER The leading dimension of the arrays A, AF, Q and R. LDA >= M.``` [in] TAU ``` TAU is REAL array, dimension (N) The scalar factors of the elementary reflectors corresponding to the QR factorization in AF.``` [out] WORK ` WORK is REAL array, dimension (LWORK)` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK.``` [out] RWORK ` RWORK is REAL array, dimension (M)` [out] RESULT ``` RESULT is REAL array, dimension (2) The test ratios: RESULT(1) = norm( R - Q'*A ) / ( M * norm(A) * EPS ) RESULT(2) = norm( I - Q'*Q ) / ( M * EPS )```

Definition at line 133 of file sqrt02.f.

135 *
136 * -- LAPACK test routine --
137 * -- LAPACK is a software package provided by Univ. of Tennessee, --
138 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
139 *
140 * .. Scalar Arguments ..
141  INTEGER K, LDA, LWORK, M, N
142 * ..
143 * .. Array Arguments ..
144  REAL A( LDA, * ), AF( LDA, * ), Q( LDA, * ),
145  \$ R( LDA, * ), RESULT( * ), RWORK( * ), TAU( * ),
146  \$ WORK( LWORK )
147 * ..
148 *
149 * =====================================================================
150 *
151 * .. Parameters ..
152  REAL ZERO, ONE
153  parameter( zero = 0.0e+0, one = 1.0e+0 )
154  REAL ROGUE
155  parameter( rogue = -1.0e+10 )
156 * ..
157 * .. Local Scalars ..
158  INTEGER INFO
159  REAL ANORM, EPS, RESID
160 * ..
161 * .. External Functions ..
162  REAL SLAMCH, SLANGE, SLANSY
163  EXTERNAL slamch, slange, slansy
164 * ..
165 * .. External Subroutines ..
166  EXTERNAL sgemm, slacpy, slaset, sorgqr, ssyrk
167 * ..
168 * .. Intrinsic Functions ..
169  INTRINSIC max, real
170 * ..
171 * .. Scalars in Common ..
172  CHARACTER*32 SRNAMT
173 * ..
174 * .. Common blocks ..
175  COMMON / srnamc / srnamt
176 * ..
177 * .. Executable Statements ..
178 *
179  eps = slamch( 'Epsilon' )
180 *
181 * Copy the first k columns of the factorization to the array Q
182 *
183  CALL slaset( 'Full', m, n, rogue, rogue, q, lda )
184  CALL slacpy( 'Lower', m-1, k, af( 2, 1 ), lda, q( 2, 1 ), lda )
185 *
186 * Generate the first n columns of the matrix Q
187 *
188  srnamt = 'SORGQR'
189  CALL sorgqr( m, n, k, q, lda, tau, work, lwork, info )
190 *
191 * Copy R(1:n,1:k)
192 *
193  CALL slaset( 'Full', n, k, zero, zero, r, lda )
194  CALL slacpy( 'Upper', n, k, af, lda, r, lda )
195 *
196 * Compute R(1:n,1:k) - Q(1:m,1:n)' * A(1:m,1:k)
197 *
198  CALL sgemm( 'Transpose', 'No transpose', n, k, m, -one, q, lda, a,
199  \$ lda, one, r, lda )
200 *
201 * Compute norm( R - Q'*A ) / ( M * norm(A) * EPS ) .
202 *
203  anorm = slange( '1', m, k, a, lda, rwork )
204  resid = slange( '1', n, k, r, lda, rwork )
205  IF( anorm.GT.zero ) THEN
206  result( 1 ) = ( ( resid / real( max( 1, m ) ) ) / anorm ) / eps
207  ELSE
208  result( 1 ) = zero
209  END IF
210 *
211 * Compute I - Q'*Q
212 *
213  CALL slaset( 'Full', n, n, zero, one, r, lda )
214  CALL ssyrk( 'Upper', 'Transpose', n, m, -one, q, lda, one, r,
215  \$ lda )
216 *
217 * Compute norm( I - Q'*Q ) / ( M * EPS ) .
218 *
219  resid = slansy( '1', 'Upper', n, r, lda, rwork )
220 *
221  result( 2 ) = ( resid / real( max( 1, m ) ) ) / eps
222 *
223  RETURN
224 *
225 * End of SQRT02
226 *
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: slaset.f:110
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine sorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
SORGQR
Definition: sorgqr.f:128
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
subroutine ssyrk(UPLO, TRANS, N, K, ALPHA, A, LDA, BETA, C, LDC)
SSYRK
Definition: ssyrk.f:169
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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