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CHEBYQAD.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CHEBYQAD
* Problem :
* *********
* The Chebyquad problem using the exact formula for the
* shifted chebyshev polynomials.
* The Hessian is full.
* Source: problem 35 in
* J.J. More', B.S. Garbow and K.E. Hillstrom,
* "Testing Unconstrained Optimization Software",
* ACM Transactions on Mathematical Software, vol. 7(1), pp. 17-41, 1981.
* See also Buckley#133 (p. 44).
* SIF input: Nick Gould, March 1990.
* classification SBR2-AN-V-0
* Number of variables
*IE N 2 $-PARAMETER
*IE N 4 $-PARAMETER
*IE N 5 $-PARAMETER
*IE N 6 $-PARAMETER
*IE N 7 $-PARAMETER
*IE N 8 $-PARAMETER
*IE N 9 $-PARAMETER
*IE N 10 $-PARAMETER original value
*IE N 20 $-PARAMETER
*IE N 50 $-PARAMETER
IE N 100 $-PARAMETER
I= M N
IA N+1 N 1
* Other parameters
IE 1 1
IE 2 2
RI RN N
RD 1/N RN 1.0
RI RN+1 N+1
RD 1/N+1 RN+1 1.0
VARIABLES
DO J 1 N
X X(J)
ND
GROUPS
DO I 1 M
XN G(I)
ND
CONSTANTS
* The constants are the (0,1) average values of the Chebyshev
* polynomials. Analytic expressions are known for these values.
* Odd averages are zero.
* Even averages are -1/(i**2 -1}
DO I 2 M
DI I 2
I* I**2 I I
IA I**2-1 I**2 -1
RI RLAST I**2-1
RD -1/LAST RLAST -1.0
Z CHEBYQAD G(I) -1/LAST
ND
BOUNDS
* defaults = variables in (0, 1)
XU CHEBYQAD 'DEFAULT' 1.0
START POINT
DO J 1 N
RI RJ J
R* START RJ 1/N+1
Z CHEBYQAD X(J) START
ND
ELEMENT TYPE
EV CHEBYPOL X
EP CHEBYPOL RI
ELEMENT USES
* The elements are the i-th (shifted) chebyshev polynomial
* evaluated at the j-th variable.
XT 'DEFAULT' CHEBYPOL
DO I 1 M
RI RI I
DO J 1 N
ZV E(I,J) X X(J)
ZP E(I,J) RI RI
ND
GROUP TYPE
GV L2 GVAR
GROUP USES
XT 'DEFAULT' L2
DO I 1 M
DO J 1 N
ZE G(I) E(I,J) 1/N
ND
OBJECT BOUND
* Solution
*LO SOLTN(2) 0.0
*LO SOLTN(4) 0.0
*LO SOLTN(5) 0.0
*LO SOLTN(6) 0.0
*LO SOLTN(7) 0.0
*LO SOLTN(8) 3.516874D-3
*LO SOLTN(9) 0.0
*LO SOLTN(10) 4.772713D-3
*LO SOLTN(20) 4.572955D-3
*LO SOLTN(50) 5.386315D-3
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CHEBYQAD
TEMPORARIES
R DIF
R Y
R SQRTY
R ACOSX
R COSAC
R SINAC
M ACOS
M COS
M SIN
M SQRT
INDIVIDUALS
* the Chebypol element, the i-th (shifted) Chebyshev polynomial
T CHEBYPOL
A DIF 2.0D+0 * X - 1.0D+0
A Y 1.0D+0 - DIF * DIF
A SQRTY SQRT( Y )
A ACOSX RI * ACOS( DIF )
A COSAC COS( ACOSX )
A SINAC SIN( ACOSX )
F COSAC
G X 2.0D+0 * RI * SINAC / SQRTY
H X X 4.0D+0 * RI * ( SINAC * DIF / SQRTY
H+ - RI * COSAC ) / Y
ENDATA
*********************
* SET UP THE GROUPS *
* ROUTINE *
*********************
GROUPS CHEBYQAD
INDIVIDUALS
T L2
F GVAR * GVAR
G GVAR + GVAR
H 2.0D+0
ENDATA