5
w0 + w1 + w2 + w3 - 1;
w0*b + w1*b + w2*b + w3*b + 1/2*w1 + w2 + 3/2*w3 - 0.63397459621556;
w0*b**2 + w1*b**2 + w2*b**2 + w3*b**2
+ w1*b + 2*w2*b + 3*w3*b + 1/4*w1 + w2 + 9/2*w3
- 0.40192378864668;
w0*b**3 + w1*b**3 + w2*b**3 + w3*b**3
+ 3/2*w1*b**2 + 3*w2*b**2 + 9/2 *w3*b**2
+ 3/4*w1*b + 3*w2*b + 27/4*w3*b
+ 1/8*w1 + w2 + 27/8*w3
- 0.13109155679036;
w0*b**4 + w1*b**4 + w2*b**4 + w3*b**4
+ 2 * w1*b**3 + 4*w2*b**3 + 6 * w3*b**3
+ 3/2* w1*b**2 + 6*w2*b**2 + 27/2* w3*b**2
+ 1/2* w1*b + 4*w2*b + 27/2* w3*b
+ 1/16*w1 + w2 + 81/16*w3
+ 0.30219332850656;
TITLE : interpolating quadrature formula for function defined on a grid
ROOT COUNTS :
total degree : 120
2-homogeneous Bezout number : 10
with partition : {w0 w1 w2 w3 }{b }
mixed volume : 10
REFERENCES :
Wim Sweldens:
`The construction and application of wavelets in numerical analysis',
PhD thesis, K.U.Leuven, 1994.
NOTES :
The general formulation of this problem is the following:
3
--- i
\ / j \
p = / w * | b + --- | - M = 0, voor i=0,..,4
i --- j \ 2 / i
j=0
with M the moments of the quadrature formula
i
Note that this system is ill-conditioned.
There are 4 complex and one real solution.
THE SOLUTIONS :
5 5
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
w0 : 5.94320041132405E-01 6.47823742644204E-01
w1 : 6.34228524314905E-01 -7.69412003529916E-01
w2 : -2.31596317854069E-01 1.14738693809794E-01
w3 : 3.04775240675963E-03 6.84956707591845E-03
b : 5.43885023302037E-01 2.59692957341287E-01
== err : 4.823E-15 = rco : 2.192E-03 = res : 2.001E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
w0 : -1.22978831272261E-01 -3.86449758136510E-01
w1 : -8.01703471950447E-01 1.30788622257006E+00
w2 : 1.67456028270032E+00 -7.66032187835804E-01
w3 : 2.50122020522384E-01 -1.55404276597748E-01
b : -1.01491698129312E+00 3.45195491447395E-01
== err : 7.626E-15 = rco : 3.397E-02 = res : 1.413E-15 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
w0 : 6.99788728057480E+00 -1.21348797918292E-59
w1 : -1.53314387602986E+01 2.30251565280862E-59
w2 : 1.05990704810461E+01 -1.33794828474015E-59
w3 : -1.26551900132224E+00 2.33363072919793E-60
b : -4.01098002697833E-01 -1.65298843318187E-61
== err : 7.602E-14 = rco : 1.345E-03 = res : 1.221E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
w0 : 5.94320041132405E-01 -6.47823742644204E-01
w1 : 6.34228524314905E-01 7.69412003529917E-01
w2 : -2.31596317854069E-01 -1.14738693809794E-01
w3 : 3.04775240675961E-03 -6.84956707591844E-03
b : 5.43885023302037E-01 -2.59692957341287E-01
== err : 3.581E-15 = rco : 2.192E-03 = res : 4.175E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
w0 : -1.22978831272260E-01 3.86449758136511E-01
w1 : -8.01703471950450E-01 -1.30788622257006E+00
w2 : 1.67456028270033E+00 7.66032187835804E-01
w3 : 2.50122020522384E-01 1.55404276597748E-01
b : -1.01491698129312E+00 -3.45195491447395E-01
== err : 7.020E-15 = rco : 3.397E-02 = res : 2.201E-15 ==