LIDAR: FUTURE OF MAPPING AND NAVIGATION
Nguyen Duy Anh
LIDAR (or more commonly now, lidar) is an acronym of Light Detection and Ranging. It was
developed in the 1960s and works similar to radar, calcu
Nanyang Technological University
School of Physical and Mathematical Sciences
Division of Mathematical Sciences
MH1811 Mathematics 2
Tutorial 6
Topics: Directional Derivatives; Global and Local maximu
Nanyang Technological University
School of Physical and Mathematical Sciences
Division of Mathematical Sciences
MH1811 Mathematics 2
Tutorial 6
In this tutorial, we will compute directional derivative
MH1811 Mathematics 2
Gradient Vectors: Orthogonality & Applications
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr
Nanyang Technological University
School of Physical and Mathematical Sciences
Division of Mathematical Sciences
MH1811 Mathematics 2
Tutorial 5
In this tutorial, we begin to explore applications of pa
MH1811 Mathematics 2
Tutorial 4
Dr Tan Geok Choo
Division of Mathematical Sciences
School of Physical and Mathematical Sciences
Nanyang Technological University
Dr Tan Geok Choo (Division of Mathemati
Sums of Independent Random Variables
Consider the sum of two independent discrete random variables X and Y
whose values are restricted to the non-negative integers. Let fX () denote the
probability di
Chapter 2. SOLUTIONS OF EQUATIONS
IN ONE VARIABLE
Bisection method (I)
f(b)
p
a
b
f(a)
Calculus: Intermediate Value Theorem Let f (x) be continuous
(f (x) C ([a, b]). If f (a)f (b) < 0, then there exi
Chapter 6. NUMERICAL INTEGRATION
Closed Newton-Cotes formulae (I)
Let us start with an example: suppose that we want to approximate
b
f (x)dx.
a
Let x0 = a and x1 = b. We have
f (x) = P1 (x) +
f (x)
(
Composite quadrature rules (I)
In the Newton-Cotes formulae with n + 1 points, we note that the
error is O(hn+2 ) when n is odd and O(hn+3 ) when n is even,
where h = (b a)/n (closed formulae) or h =
Chapter 4. NUMERICAL DIFFERENTIATION
Numerical Dierentiation (I)
In this chapter, we will learn how to approximate derivatives of a
function.
The most well known example:
From the denition:
lim
h0
f
Chapter 5. RICHARDSONS EXTRAPOLATION
Richardsons extrapolation (I)
Suppose that we have an approximation with the error O(h),
how can we get a better approximation with the error O(h2 )?
Richardsons
Divided dierences (I)
Lets come back to our example:
Suppose we know that at x0 = 1, f (x0 ) = 1 and at x1 = 2,
f (x1 ) = 3/2.
The interpolating polynomial we found is
1
P1 (x) = (x + 1).
2
As we did
Chapter 3. INTERPOLATION
What is interpolation?
In many applications, we collect data at only some points, for
example, at some moments, or at some spatial points.
We thus only know the values of a
Newton method (I)
For solving an equation f (x) = 0, sometimes it is better to nd a
function g (x) so that xed points of g are the roots of f (x) = 0
and nd xed points of g .
There are many such fun
CHAPTER 1. INTRODUCTION
Dierentiation (I)
Product rule
(fg ) = f g + g f .
so
(fgh) = f (gh) + (gh) f = f gh + g hf + h gf .
Example
d
(x x1 )(x x2 )(x x3 ) = (x x2 )(x x3 ) + (x x1 )(x x3 )
dx
+(x x
Numerical Analysis I. Tutorial and laboratory 12
1. TUTORIAL PROBLEMS
1. Suppose that f (0) = 1, f (0.5) = 2.5, f (1) = 2 and f (0.25) = f (0.75) =
. Find if the composite trapezoidal rule with n = 4
Numerical Analysis I. Tutorial and Laboratory 10
1. TUTORIAL PROBLEMS
2
1. (a) The trapezoidal rule applied to 0 f (x)dx gives the value 4, and the
Simpsons rule gives the value 2. What is f (1).
2
(b
Numerical Analysis I. Tutorial and laboratory 11
1. TUTORIAL PROBLEMS
1. Find a, b, c and d so that the quadrature formula
1
f (x)dx af (1) + bf (1) + cf (1) + df (1)
1
has degree of precision 3.
2. V
Numerical Analysis I. Tutorial and Laboratory 1
1. TUTORIAL PROBLEMS
1. Let |h| < 1. Which of the following functions are O(h)? Explain why.
(a) 2h, (b) 4h, (c) h+h2 , (d) 3h+h3 , (e) |h|1/2 , (f) |h|
Numerical Analysis I. Tutorial 9
1. Suppose that N (h) is an approximation for M for every h > 0 and
that
M = N (h) + K1 h + K2 h2 + K3 h3 + . . . ,
for some constants K1 , K2 , K3 , . . . Use the val
Numerical Analysis I. Tutorial and Laboratory 8
1. TUTORIAL PROBLEMS
1. Derive the approximation
f (a) =
f (a + h) 2f (a) + f (a h)
+ O(h2 )
2
h
from Lagrange interpolation and from Taylor expansion.