LAPACK  3.8.0 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' A U - S | / ( |A| m ulp ) *andC>              RESULT(2) = | I - U'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' | / ( |A| n ulp )   *andC>             RESULT(2) = | I - UU' | / ( 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)' 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.```
Date
December 2016

Definition at line 157 of file dsyt22.f.

157 *
158 * -- LAPACK test routine (version 3.7.0) --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 * December 2016
162 *
163 * .. Scalar Arguments ..
164  CHARACTER uplo
165  INTEGER itype, kband, lda, ldu, ldv, m, n
166 * ..
167 * .. Array Arguments ..
168  DOUBLE PRECISION a( lda, * ), d( * ), e( * ), result( 2 ),
169  \$ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
170 * ..
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175  DOUBLE PRECISION zero, one
176  parameter( zero = 0.0d0, one = 1.0d0 )
177 * ..
178 * .. Local Scalars ..
179  INTEGER j, jj, jj1, jj2, nn, nnp1
180  DOUBLE PRECISION anorm, ulp, unfl, wnorm
181 * ..
182 * .. External Functions ..
183  DOUBLE PRECISION dlamch, dlansy
184  EXTERNAL dlamch, dlansy
185 * ..
186 * .. External Subroutines ..
187  EXTERNAL dgemm, dort01, dsymm
188 * ..
189 * .. Intrinsic Functions ..
190  INTRINSIC dble, max, min
191 * ..
192 * .. Executable Statements ..
193 *
194  result( 1 ) = zero
195  result( 2 ) = zero
196  IF( n.LE.0 .OR. m.LE.0 )
197  \$ RETURN
198 *
199  unfl = dlamch( 'Safe minimum' )
200  ulp = dlamch( 'Precision' )
201 *
202 * Do Test 1
203 *
204 * Norm of A:
205 *
206  anorm = max( dlansy( '1', uplo, n, a, lda, work ), unfl )
207 *
208 * Compute error matrix:
209 *
210 * ITYPE=1: error = U' A U - S
211 *
212  CALL dsymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
213  nn = n*n
214  nnp1 = nn + 1
215  CALL dgemm( 'T', 'N', m, m, n, one, u, ldu, work, n, zero,
216  \$ work( nnp1 ), n )
217  DO 10 j = 1, m
218  jj = nn + ( j-1 )*n + j
219  work( jj ) = work( jj ) - d( j )
220  10 CONTINUE
221  IF( kband.EQ.1 .AND. n.GT.1 ) THEN
222  DO 20 j = 2, m
223  jj1 = nn + ( j-1 )*n + j - 1
224  jj2 = nn + ( j-2 )*n + j
225  work( jj1 ) = work( jj1 ) - e( j-1 )
226  work( jj2 ) = work( jj2 ) - e( j-1 )
227  20 CONTINUE
228  END IF
229  wnorm = dlansy( '1', uplo, m, work( nnp1 ), n, work( 1 ) )
230 *
231  IF( anorm.GT.wnorm ) THEN
232  result( 1 ) = ( wnorm / anorm ) / ( m*ulp )
233  ELSE
234  IF( anorm.LT.one ) THEN
235  result( 1 ) = ( min( wnorm, m*anorm ) / anorm ) / ( m*ulp )
236  ELSE
237  result( 1 ) = min( wnorm / anorm, dble( m ) ) / ( m*ulp )
238  END IF
239  END IF
240 *
241 * Do Test 2
242 *
243 * Compute U'U - I
244 *
245  IF( itype.EQ.1 )
246  \$ CALL dort01( 'Columns', n, m, u, ldu, work, 2*n*n,
247  \$ result( 2 ) )
248 *
249  RETURN
250 *
251 * End of DSYT22
252 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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, or the element of largest absolute value of a real symmetric matrix.
Definition: dlansy.f:124
subroutine dsymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DSYMM
Definition: dsymm.f:191
subroutine dgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
DGEMM
Definition: dgemm.f:189
subroutine dort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
DORT01
Definition: dort01.f:118
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