Quadrature Formulae. Algebraic Degree of Precision of Quadrature Formulas.
Newton-Cotes Quadratures. Composite Newton-Cotes Quadratures. EulerMacLaurin Quadrature Formula.
Topics 10 and 11 according to the Outline
What is a Quadrature Formula? Quadrature
MATH 354 and MAST 334, Section A, Assignment 11, Solutions
Problem R1. By using an appropriate composite Simpson quadrature formula approximate the exact
value I = 0.5 sin(x2 )dx with 6 true (significant) digits. How many nodes you have used to ensure
MATH-354, MAST-334, Numerical Analysis, Fall Semester, 2015
Assignment 8, Solutions
Problem 3a/ p.153. Construct the free cubic spline for the following data:
Solution. The unique free cubic spline interpolant on any two
4. Error Analysis for Iterative Fixed-Point Methods. Order of Convergence.
Accelerating Convergence - Aitkens Accelerating Procedure, Steffensens Accelerating Method.
Error Analysis. The main goal of the present lecture is to classify (to order) the conve
MATH 354 and MAST 334, Fall 2015
Assignment 5, Solutions
Problem 1/p.85 (modified). (a) Determine the exact order of convergence of the
Newton-Raphson fixed-point method to approximate the solution p [0, 1] of the nonlinear equation
x2 2xex + e2x = 0 for
MATH-354/ MAST-334, Fall 2015
Assignment 1, Solutions
Problem 11, page 29. Let
f (x) =
x cos(x) sin(x)
lim f (x).
(b) Use four-digit rounding arithmetic to evaluate f (0.1). In other words, find four-digit
rounding approximation to
MATH-354 / MAST-334, Fall 2015
Assignment 2, Solutions
Problem 11, page 15. (a) Find the third Taylor polynomial P3 (x) for the function
f (x) = (x 1) ln(x) about x0 = 1.
(b) Use P3 (0.5) to approximate f (0.5). Find an upper bound for the error |f (0.5)P
MATH-354, MAST-334, Section A, Fall 2015
Assignment 4, Solutions
Problem 6,(a), 8,(a), page 75 (modified). Find approximations accurate to within
108 to the exact solutions of the non-linear equation
ex + 2x + 2 cos(x) 6 = 0
located in the interval [1, 2]
MATH-354, MAST-334, Section A
Assignment 3, Solutions
Problem 2, (a),(b),(d); page 29. Locate all solutions of the given non-linear equation
in intervals of length at most 0.5.
x 3x = 0;
(b) 3x2 e0.5x = 0;
(d) x3 + 4.001x2 + 4.002x + 1.101 = 0
Numerical Analysis, MATH 354 and MAST 334, Section A
Numerical Analysis by Dr. Lloyd Nicholas Trefethen, professor of numerical
analysis and head of the Numerical Analysis Group at the Mathematical Institute, University of Oxford. . Numerical Analysis has
MATH354/Mast334 Assignment 3
Section 2.3, p.75
1. N 6(b).
With Maple procedure p0 = 1.5, p1 = 1.378706774, p2 = 1.397135813,
p3 = 1.397747837, p4 = 1.397748476. Hence, required number of iterations is n = 4.
2. N 8(b).
With Maple procedure p0 =
MATH354/Mast334 Assignment 2
Section 2.1, p.
1. N 8.
(b) From the graph it is easy to see that the root appears near x = 4.5.
For f (x) = x tan(x) f (4.4) = 1.393676219 > 0 and f (4.6) =
4.260174896 < 0, and it follows from continuity of f a
MATH-354, MAST-334, Numerical Analysis, Fall 2015
Assignment 6, Solutions
Problem 1,c, 3,c/pages 114-115. (a) For f (x) = ln(x + 1) and x0 = 0, x1 = 0.6,
x2 = 0.9 construct Lagrange interpolating polynomials P1 (x) of degree at most one, and
P2 (x) of deg
12. Gaussian Quadrature Formula.
Quadratures of Maximal Polynomial Degree of Precision.
Interpolatory Quadrature Formula with Weight Function. Let us approximate
f (x) defined on [1, 1] by using n distinct interpolating nodes x1 , x2 , . . . , xn with Lag
Week 5. Topics: Lagrange Interpolating Polynomial. Interpolation error.
Nevilles Algorithm. Inverse Lagrange interpolation to approximate solutions
of nonlinear equations. Chebyshev Polynomials. Minimization of the Interpolation Error. Piece-wise linear I
Numerical Analysis, MATH 354 and MAST 334
4.1 Numerical Differentiation. 4.2 Richardsons Extrapolation.
4.1. Numerical Differentiation
The derivative of a function f (x) at x = x0 is
f (x0 ) = lim
f (x0 + h) f (x0 )
The above formula gives an o
MATH 354 and MAST 334, Fall 2105
4.5 Romberg Integration. Construction of Higher Order Convergent Quadrature Formulae.
Composite Trapezoidal Rule (Quadrature Formula) on n sub-intervals, using (n + 1)
xi = a + i
, i = 0, 1, . . . , n; x
2. Numerical Solution of Nonlinear Equations. Location of Solutions of Nonlinear Equations in Intervals. Bisectional Method. Fixed-Point Method.
Location of Solutions of Nonlinear Equations in Intervals. In this section we shall
need the following 2 theor
8.6 Interpolation by Trigonometric Polynomials.
In Section 8.5 we computed the coefficients of the best discrete least squares trigonometric polynomial
approximant Sn (x) Tn,c for a function f (x) on the interval [, ] based on 2m data cfw_(xj , yj )2m1
In [0,1] the polynomial is concave, in [2,2.5] it is convex. Hence N-R method with p0=1 will be
convergent to the solution in [0,1] and with p0=2.5 N-R method will be convergent to the solution
in [2,2.5]. The initial iteration should satisfy f(p0)*f^
Problem 1. Find the continuous least squares trigonometric approximant s(x) 2 T2,c to
the function f(x)=sin(
in the intertval [0,1].
Draw the graphs of f(x), s(x) on one plot and next, use another plot to draw the absolute
approximation error |f(x)-s(x)|
(1) Definition. Suppose that p is a solution of f (x) = 0. We say that p is a solution of
f (x) = 0 of multiplicity m-positive integer if
f (x) = (x p)m g(x)
g(p) 6= 0.
or equivalently by using Taylors Polynomial Approximation Theorem,
f (p) = 0,
MATH 354 and MAST 334, Assignment 10, Solutions
Problem 1. (a) By using the method of undetermined coefficients derive a first-derivative, 5data midpoint (central difference) formula for numerical differentiation based on the data f (x0
2h), f (x0 h), f
Section 1.1, p.14
1. N 2(b).
From the graph we see that there are three roots, one negative and two
positive. Denote f (x) = 4x2 ex . Since f 0 (x) = 8x ex , then there is a