```
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 ==
```