LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ ssyt22()

 subroutine ssyt22 ( integer ITYPE, character UPLO, integer N, integer M, integer KBAND, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( * ) TAU, real, dimension( * ) WORK, real, dimension( 2 ) RESULT )

SSYT22

Purpose:
```      SSYT22  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, SSYT22 does nothing.
It must be at least zero.
Not modified.

M       INTEGER
The number of columns of U.  If it is zero, SSYT22 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       REAL 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       REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
Not modified.

E       REAL 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       REAL 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       REAL 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     REAL 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    REAL array, dimension (2*N**2)
Workspace.
Modified.

RESULT  REAL 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 ssyt22.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  REAL A( LDA, * ), D( * ), E( * ), RESULT( 2 ),
168  \$ TAU( * ), U( LDU, * ), V( LDV, * ), WORK( * )
169 * ..
170 *
171 * =====================================================================
172 *
173 * .. Parameters ..
174  REAL ZERO, ONE
175  parameter( zero = 0.0e0, one = 1.0e0 )
176 * ..
177 * .. Local Scalars ..
178  INTEGER J, JJ, JJ1, JJ2, NN, NNP1
179  REAL ANORM, ULP, UNFL, WNORM
180 * ..
181 * .. External Functions ..
182  REAL SLAMCH, SLANSY
183  EXTERNAL slamch, slansy
184 * ..
185 * .. External Subroutines ..
186  EXTERNAL sgemm, ssymm
187 * ..
188 * .. Intrinsic Functions ..
189  INTRINSIC max, min, real
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 = slamch( 'Safe minimum' )
199  ulp = slamch( 'Precision' )
200 *
201 * Do Test 1
202 *
203 * Norm of A:
204 *
205  anorm = max( slansy( '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 ssymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
212  nn = n*n
213  nnp1 = nn + 1
214  CALL sgemm( '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 = slansy( '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, real( 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 sort01( 'Columns', n, m, u, ldu, work, 2*n*n,
246  \$ result( 2 ) )
247 *
248  RETURN
249 *
250 * End of SSYT22
251 *
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 ssymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SSYMM
Definition: ssymm.f:189
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
subroutine sort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
SORT01
Definition: sort01.f:116
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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