LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 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' 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, 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)' 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.```
Date
November 2011

Definition at line 157 of file ssyt22.f.

157 *
158 * -- LAPACK test routine (version 3.4.0) --
159 * -- LAPACK is a software package provided by Univ. of Tennessee, --
160 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
161 * November 2011
162 *
163 * .. Scalar Arguments ..
164  CHARACTER uplo
165  INTEGER itype, kband, lda, ldu, ldv, m, n
166 * ..
167 * .. Array Arguments ..
168  REAL a( lda, * ), d( * ), e( * ), result( 2 ),
169  \$ tau( * ), u( ldu, * ), v( ldv, * ), work( * )
170 * ..
171 *
172 * =====================================================================
173 *
174 * .. Parameters ..
175  REAL zero, one
176  parameter ( zero = 0.0e0, one = 1.0e0 )
177 * ..
178 * .. Local Scalars ..
179  INTEGER j, jj, jj1, jj2, nn, nnp1
180  REAL anorm, ulp, unfl, wnorm
181 * ..
182 * .. External Functions ..
183  REAL slamch, slansy
184  EXTERNAL slamch, slansy
185 * ..
186 * .. External Subroutines ..
187  EXTERNAL sgemm, ssymm
188 * ..
189 * .. Intrinsic Functions ..
190  INTRINSIC max, min, real
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 = slamch( 'Safe minimum' )
200  ulp = slamch( 'Precision' )
201 *
202 * Do Test 1
203 *
204 * Norm of A:
205 *
206  anorm = max( slansy( '1', uplo, n, a, lda, work ), unfl )
207 *
208 * Compute error matrix:
209 *
210 * ITYPE=1: error = U' A U - S
211 *
212  CALL ssymm( 'L', uplo, n, m, one, a, lda, u, ldu, zero, work, n )
213  nn = n*n
214  nnp1 = nn + 1
215  CALL sgemm( '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 = slansy( '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, REAL( 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 sort01( 'Columns', n, m, u, ldu, work, 2*n*n,
247  \$ result( 2 ) )
248 *
249  RETURN
250 *
251 * End of SSYT22
252 *
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:189
subroutine sort01(ROWCOL, M, N, U, LDU, WORK, LWORK, RESID)
SORT01
Definition: sort01.f:118
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine ssymm(SIDE, UPLO, M, N, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SSYMM
Definition: ssymm.f:191
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, or the element of largest absolute value of a real symmetric matrix.
Definition: slansy.f:124

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