LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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.```

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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