# Mathematical Experiment 8

1. Let f be an arbitrary function. For any two distinct points x0,x1 set
2.  [ x0x1] = f(x0)-f(x1) x0-x1 .
For any three distinct points x0,x1,x2 set
 [ x0x1x2] = [ x0x1] -[x1x2] x0-x2 .
In general if x0,x1,¼,xn are distinct, set
 [ x0x1x2¼xn] = [x0x1x2¼xn-1] -[ x1x2x3¼xn] x0-xn .
It is a matter of algebraic substitution to verify that for a polynomial f(x) of degree n the identity
 f(x)
 =
 f(x0)+[ x0x1] (x-x0)+[x0x1x2] (x-x0)(x-x1)
 +¼+[ x0x1x2¼xn](x-x0)(x-x1)¼(x-xn-1)
holds for any distinct x0,x1,x2,¼,xn. The spreadsheet can then be applied to solve the polynomial interpolation problem based on this formula of divided differences. The basic procedure is to set up a table to compute the divided differences according to this scheme:

3. Draw the graph of the interpolation polynomial satisfying the following condition with Maple:
4.  x 3 2 6 5 1 f(x) 3 -3 1 4 2
5. When the package ``networks'' is loaded, a number of new commands are added. The most important one is draw. This allows you to visualize the graph.
• Commands for creating graphs: complete, cube, dodecahedron, graph, new, octahedron, icosahedron. petersen, random, tetrahedron, void,
• Command used to display properties of a graph: show.
• Information about a graph: arrivals, departures, edges, ends, eweight, getlabel, head, incident, neighbors, tail, vertices, vweight.
• Computations in terms of polynomials: acycpoly, charpoly, chrompoly, flowpoly, rankpoly, spanpoly, tuttepoly.
• Create new graph from an old one: addedge, addvertex, complement, connect, contract,delete, duplicate, gsimp, gunion, induce, shrink.
• Computation associated with a graph: allpairs, bicomponents, components, connectivity, countcuts, counttrees, diameter, djspantree, girth, isplanar, path, rank, shortpathtree, span, spantree.
• Degree: degreeseq, indegree, mindegree, maxdegree, outdegree, vdegree.
• Directed tree: ancestor, daughter.
• Cycle: cycle, cyclebase, fundcyc.
• Network: dinic, flow, mincut.