LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dsyt22()

subroutine dsyt22 ( integer  ITYPE,
character  UPLO,
integer  N,
integer  M,
integer  KBAND,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( * )  D,
double precision, dimension( * )  E,
double precision, dimension( ldu, * )  U,
integer  LDU,
double precision, dimension( ldv, * )  V,
integer  LDV,
double precision, dimension( * )  TAU,
double precision, dimension( * )  WORK,
double precision, dimension( 2 )  RESULT 
)

DSYT22

Purpose:
      DSYT22  generally checks a decomposition of the form

              A U = U S

      where A is symmetric, the columns of U are orthonormal, and S
      is diagonal (if KBAND=0) or symmetric tridiagonal (if
      KBAND=1).  If ITYPE=1, then U is represented as a dense matrix,
      otherwise the U is expressed as a product of Householder
      transformations, whose vectors are stored in the array "V" and
      whose scaling constants are in "TAU"; we shall use the letter
      "V" to refer to the product of Householder transformations
      (which should be equal to U).

      Specifically, if ITYPE=1, then:

              RESULT(1) = | U**T A U - S | / ( |A| m ulp ) and
              RESULT(2) = | I - U**T U | / ( m ulp )
  ITYPE   INTEGER
          Specifies the type of tests to be performed.
          1: U expressed as a dense orthogonal matrix:
             RESULT(1) = | A - U S U**T | / ( |A| n ulp )  and
             RESULT(2) = | I - U U**T | / ( n ulp )

  UPLO    CHARACTER
          If UPLO='U', the upper triangle of A will be used and the
          (strictly) lower triangle will not be referenced.  If
          UPLO='L', the lower triangle of A will be used and the
          (strictly) upper triangle will not be referenced.
          Not modified.

  N       INTEGER
          The size of the matrix.  If it is zero, DSYT22 does nothing.
          It must be at least zero.
          Not modified.

  M       INTEGER
          The number of columns of U.  If it is zero, DSYT22 does
          nothing.  It must be at least zero.
          Not modified.

  KBAND   INTEGER
          The bandwidth of the matrix.  It may only be zero or one.
          If zero, then S is diagonal, and E is not referenced.  If
          one, then S is symmetric tri-diagonal.
          Not modified.

  A       DOUBLE PRECISION array, dimension (LDA , N)
          The original (unfactored) matrix.  It is assumed to be
          symmetric, and only the upper (UPLO='U') or only the lower
          (UPLO='L') will be referenced.
          Not modified.

  LDA     INTEGER
          The leading dimension of A.  It must be at least 1
          and at least N.
          Not modified.

  D       DOUBLE PRECISION array, dimension (N)
          The diagonal of the (symmetric tri-) diagonal matrix.
          Not modified.

  E       DOUBLE PRECISION array, dimension (N)
          The off-diagonal of the (symmetric tri-) diagonal matrix.
          E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
          Not referenced if KBAND=0.
          Not modified.

  U       DOUBLE PRECISION array, dimension (LDU, N)
          If ITYPE=1 or 3, this contains the orthogonal matrix in
          the decomposition, expressed as a dense matrix.  If ITYPE=2,
          then it is not referenced.
          Not modified.

  LDU     INTEGER
          The leading dimension of U.  LDU must be at least N and
          at least 1.
          Not modified.

  V       DOUBLE PRECISION array, dimension (LDV, N)
          If ITYPE=2 or 3, the lower triangle of this array contains
          the Householder vectors used to describe the orthogonal
          matrix in the decomposition.  If ITYPE=1, then it is not
          referenced.
          Not modified.

  LDV     INTEGER
          The leading dimension of V.  LDV must be at least N and
          at least 1.
          Not modified.

  TAU     DOUBLE PRECISION array, dimension (N)
          If ITYPE >= 2, then TAU(j) is the scalar factor of
          v(j) v(j)**T in the Householder transformation H(j) of
          the product  U = H(1)...H(n-2)
          If ITYPE < 2, then TAU is not referenced.
          Not modified.

  WORK    DOUBLE PRECISION array, dimension (2*N**2)
          Workspace.
          Modified.

  RESULT  DOUBLE PRECISION array, dimension (2)
          The values computed by the two tests described above.  The
          values are currently limited to 1/ulp, to avoid overflow.
          RESULT(1) is always modified.  RESULT(2) is modified only
          if LDU is at least N.
          Modified.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 155 of file dsyt22.f.

157*
158* -- LAPACK test routine --
159* -- LAPACK is a software package provided by Univ. of Tennessee, --
160* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161*
162* .. Scalar Arguments ..
163 CHARACTER UPLO
164 INTEGER ITYPE, KBAND, LDA, LDU, LDV, M, N
165* ..
166* .. Array Arguments ..
167 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
168 $ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
169* ..
170*
171* =====================================================================
172*
173* .. Parameters ..
174 DOUBLE PRECISION ZERO, ONE
175 parameter( zero = 0.0d0, one = 1.0d0 )
176* ..
177* .. Local Scalars ..
178 INTEGER J, JJ, JJ1, JJ2, NN, NNP1
179 DOUBLE PRECISION ANORM, ULP, UNFL, WNORM
180* ..
181* .. External Functions ..
182 DOUBLE PRECISION DLAMCH, DLANSY
183 EXTERNAL dlamch, dlansy
184* ..
185* .. External Subroutines ..
186 EXTERNAL dgemm, dort01, dsymm
187* ..
188* .. Intrinsic Functions ..
189 INTRINSIC dble, max, min
190* ..
191* .. Executable Statements ..
192*
193 result( 1 ) = zero
194 result( 2 ) = zero
195 IF( n.LE.0 .OR. m.LE.0 )
196 $ RETURN
197*
198 unfl = dlamch( 'Safe minimum' )
199 ulp = dlamch( 'Precision' )
200*
201* Do Test 1
202*
203* Norm of A:
204*
205 anorm = max( dlansy( '1', uplo, n, a, lda, work ), unfl )
206*
207* Compute error matrix:
208*
209* ITYPE=1: error = U**T A U - S
210*
211 CALL dsymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
212 nn = n*n
213 nnp1 = nn + 1
214 CALL dgemm( 'T', 'N', m, m, n, one, u, ldu, work, n, zero,
215 $ work( nnp1 ), n )
216 DO 10 j = 1, m
217 jj = nn + ( j-1 )*n + j
218 work( jj ) = work( jj ) - d( j )
219 10 CONTINUE
220 IF( kband.EQ.1 .AND. n.GT.1 ) THEN
221 DO 20 j = 2, m
222 jj1 = nn + ( j-1 )*n + j - 1
223 jj2 = nn + ( j-2 )*n + j
224 work( jj1 ) = work( jj1 ) - e( j-1 )
225 work( jj2 ) = work( jj2 ) - e( j-1 )
226 20 CONTINUE
227 END IF
228 wnorm = dlansy( '1', uplo, m, work( nnp1 ), n, work( 1 ) )
229*
230 IF( anorm.GT.wnorm ) THEN
231 result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
232 ELSE
233 IF( anorm.LT.one ) THEN
234 result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
235 ELSE
236 result( 1 ) = min( wnorm / anorm, dble( m ) ) / ( m*ulp )
237 END IF
238 END IF
239*
240* Do Test 2
241*
242* Compute U**T U - I
243*
244 IF( itype.EQ.1 )
245 $ CALL dort01( 'Columns', n, m, u, ldu, work, 2*n*n,
246 $ result( 2 ) )
247*
248 RETURN
249*
250* End of DSYT22
251*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine dsymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYMM
Definition: dsymm.f:189
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:187
subroutine dort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
DORT01
Definition: dort01.f:116
double precision function dlansy(NORM, UPLO, N, A, LDA, WORK)
DLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: dlansy.f:122
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