Excel Definitions

Polynomials

• Example 7: Estimate the definite integral -22 (x5+2x4-3x2+7x+12)dx.
• Choose equal subinterval widths x = (b-a)/n and sample points xi* to be each subinterval’s midpoint, i.e. xi*=a+(2*i-1)*(b-a)/2n, for a=-2, b=2, and n=100.
• Do this first with the polynomial’s formula.
• a=-2;
• b=2;
• n=100;
• dx = (b-a)/n;
• xstar = a+dx/2:dx:b-dx/2;
• Rn = sum(xstar.^5+2*xstar.^4-3*xstar.^2+7*xstar+12)*dx
• Repeat with “polyval”.
• Rn = sum(polyval(p,xstar))*dx
• In each case, you should get the approximation to be about 57.5931.
• What is the actual value of this integral?
• Answer: 288/5.

Polynomials – polyder

• Using the command “polyder”, we can differentiate a polynomial, polynonial product, or polynomial quotient!
• Syntax:
• “polyder(p)” returns the derivative of the polynomial whose coefficients are the elements of vector p.
• “[K] = polyder(a,b)” returns the derivative of polynomial a*b.
• “[Q,D]” = polyder(b,a) returns the derivative of the polynomial ratio b/a, represented as Q/D.

• Try this command with
• p = [1 2 0 -3 7 12]
• a = [1 1]
• b = [1 2 1]
• Do your results agree with calculations done by hand?

Polynomials – polyint

• Using the command “polyint”, we can integrate a polynomial!
• Syntax:
• “polyint(p,K)” returns a polynomial representing the integral of polynomial p, using a scalar constant of integration K.
• “polyint(p)” assumes a constant of integration K=0.

• Try this command with
• p = [1 2 0 -3 7 12]
• K = 7
• Do your results agree with calculations done by hand?

Fitting Polynomials to Data

• We can use the command “polyfit” to fit a polynomial to a set of data via a least-squares fit.
• Syntax: p = polyfit(x, y, n) will fit a polynomial of degree n to data given as ordered pairs (xi,yi) for i = 1, 2, … , m.