- A very simple application of the
- Show that the solution to M(z)X(z)=1 mod(z-a) is X(z)=1/M(a), provided M(a) isn't 0.

This is useful for the polynomial Chinese remainder theorem solution procedure. - Show you can formulate the interpolation problem X(z
_{i})=c_{i}as: X(z)=c_{i}mod(z-z_{i}), i=1...n. - Apply the Chinese remainder theorem to derive the Lagrange interpolation formula.

*polynomial Chinese remainder theorem*:

- Show that the solution to M(z)X(z)=1 mod(z-a) is X(z)=1/M(a), provided M(a) isn't 0.
- Apply
*Winograd*to derive a fast algorithm for multiplying quadratic polynomials mod(z³ -8):

y_{0}+y_{1}z+y_{2}z²= (h_{0}+h_{1}z+h_{2}z²) (u_{0}+u_{1}z+u_{2}z²) mod(z³ -8).

HINTS: (1) z^{3}-8=(z-2)(z² +2z+4); (2) z² +2z+4=(z-2)(z+4)+12.

You can use the latter hint to avoid the Euclidian algorithm for polynomials, if desired.

- We wish to multiply (3458)(2992) using
- Compute the residues of 3458 and 2992 for each modulus (total of 16 numbers).
- Compute the residues of the product for each modulus (total of 8 numbers).
- Use the Chinese remainder theorem to compute (3458)(2992). Confirm this is right.

*residue number systems*(illustrative example).

Use as moduli: 2,3,5,7,11,13,17,19; their product is about 10 million (large enough).

*Blind deconvolution of integer sequences*(useful for sequences scaled to integers):

We observe y_{n}=h_{n}*u_{n}={143,96,199,97,217,199,274,156,146,114,78,83,34,11,1}.

GOAL: To reconstruct BOTH h_{n}and u_{n}from their convolution y_{n}(2 unknowns).

All we know is h_{n}and u_{n}are scaled to positive integers, 0 < h,u < 10, n > 0.

Compute h_{n}and u_{n}. HINT: Use the Euclidian algorithm repeatedly, and 143=(13)(11).

*Multichannel blind deconvolution*(this is presently a "hot" topic; see journals):

We observe y_{n}¹=h_{n}*u_{n}¹={1,13,78,286,715,1287,1716,1716,1287,715,286,78,13,1}

and observe y_{n}²=h_{n}*u_{n}²={1,7,23,49,80,112,144,176,199,185,121,47,8}.

GOAL: Compute h_{n},u_{n}¹,u_{n}² from y_{n}¹,y_{n}².

These sequences need NOT be integer-valued; we know NOTHING except that WLOG h_{0}=1.

- Compute h
- By finding the
*common zeros*of Y¹(z) and Y²(z), and using them to compute H(z).

Don't be surprised if this doesn't work well; how do you identify exactly common zeros? - By applying the
*Euclidian algorithm for polynomials*to Y¹(z) and Y²(z).

- By writing Y¹(z)U²(z)-Y²(z)U¹(z)=0 as the linear system of equations

_{n}, u_{n}¹ and u_{n}² in three ways (which one is easiest?):

- By finding the