7.2.3 Chebyshev Polynomials An Nth-order polynomial in the variable x is defined as PN (x) = anxN + aN-1xN-1+ …+a1x + ao, aN # 0, (7.56) where the a; (for i = 0,1, 2, ..., N) are constants, and N is a nonnegative integer. For example, the following polynomials are, respectively, of zero, first, and third order: Po(x) = 3, P1 (x) = x – 1, P3(x) = 5x³ + x² – 2x + 4. (7.57) A number of special classes of polynomials are of great value for the inves- tigation of certain problems that arise in both pure and applied mathematics. In particular, it can be shown that any "reasonable" function of æ, defined over the interval –1< x < +1, can be written as a sum of the members of any one of these special classes of polynomials. An example is the class of power functions whose Nth member is PN = xN. Hence, any (reasonable) function f (x) has the representation f (x) = boPo(x) + bịPa (x)+ · · ·+ bN PN (x) + · · · (7.58) m=0 The series defined by the second line on the right side of equation (7.58) is known as a Taylor series. In general, the series expansion of an arbitrary function contains an unlimited (infinite) number of terms. An important example of such a class of polynomial functions is the Cheby- shev polynomials denoted by the symbol Tk(x). They are defined by the re- currence formula Tk+2 – xTr+1 + T = 0, (7.59) where |æ| < 1, and To = 2, Tị = x. In this section, we will investigate certain properties of these functions. Using the above equation and the given values for To and T1, the first several Chebyshev polynomials can be easily calculated; they are 1 T2(x) = x² 2' 3x T2(x) = x³ (7.60) 4 1 - 22 + T4(x) = x4 Proceeding in this way, we can obtain T(æ) for any finite integer k. How- ever, this procedure is very laborious and it would be much better to have Diff
7.2.3 Chebyshev Polynomials An Nth-order polynomial in the variable x is defined as PN (x) = anxN + aN-1xN-1+ …+a1x + ao, aN # 0, (7.56) where the a; (for i = 0,1, 2, ..., N) are constants, and N is a nonnegative integer. For example, the following polynomials are, respectively, of zero, first, and third order: Po(x) = 3, P1 (x) = x – 1, P3(x) = 5x³ + x² – 2x + 4. (7.57) A number of special classes of polynomials are of great value for the inves- tigation of certain problems that arise in both pure and applied mathematics. In particular, it can be shown that any "reasonable" function of æ, defined over the interval –1< x < +1, can be written as a sum of the members of any one of these special classes of polynomials. An example is the class of power functions whose Nth member is PN = xN. Hence, any (reasonable) function f (x) has the representation f (x) = boPo(x) + bịPa (x)+ · · ·+ bN PN (x) + · · · (7.58) m=0 The series defined by the second line on the right side of equation (7.58) is known as a Taylor series. In general, the series expansion of an arbitrary function contains an unlimited (infinite) number of terms. An important example of such a class of polynomial functions is the Cheby- shev polynomials denoted by the symbol Tk(x). They are defined by the re- currence formula Tk+2 – xTr+1 + T = 0, (7.59) where |æ| < 1, and To = 2, Tị = x. In this section, we will investigate certain properties of these functions. Using the above equation and the given values for To and T1, the first several Chebyshev polynomials can be easily calculated; they are 1 T2(x) = x² 2' 3x T2(x) = x³ (7.60) 4 1 - 22 + T4(x) = x4 Proceeding in this way, we can obtain T(æ) for any finite integer k. How- ever, this procedure is very laborious and it would be much better to have Diff
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Topic Video
Question
Explain the determine blue and the eqaution is here
Transcribed Image Text:An important example of such a class of polynomial functions is the Cheby-
shev polynomials denoted by the symbol T(x). They are defined by the re-
currence formula
()
Tk+2
xTR+1 +
Tk = 0,
4
(7.59)
![7.2.3
Chebyshev Polynomials
An Nth-order polynomial in the variable is defined as
PN (x) = .
anaN
+ aN-12N-1
+...+ a1x + ao, aN + 0,
(7.56)
where the a; (for i = 0,1, 2,..., N) are constants, and N is a nonnegative
integer. For example, the following polynomials are, respectively, of zero, first,
and third order:
Po(x) = 3,
P1 (x) = x – 1,
P3(x) = 5x3 + x² – 2x + 4.
(7.57)
A number of special classes of polynomials are of great value for the inves-
tigation of certain problems that arise in both pure and applied mathematics.
In particular, it can be shown that any "reasonable" function of x, defined
over the interval –1 <x < +1, can be written as a sum of the members of any
one of these special classes of polynomials. An example is the class of power
functions whose Nth member is PN = xN. Hence, any (reasonable) function
f (x) has the representation
f(x) = boPo(x) + 61P1 (x) + · · ·+ bN PN (x) + · ..
(7.58)
E bmam.
m=0
The series defined by the second line on the right side of equation (7.58) is
known as a Taylor series. In general, the series expansion of an arbitrary
function contains an unlimited (infinite) number of terms.
An important example of such a class of polynomial functions is the Cheby-
shev polynomials denoted by the symbol T(x). They are defined by the re-
currence formula
() .
Tk-+2 – xT+1 +
Tk = 0,
(7.59)
where |x| < 1, and To = 2, T1 = x. In this section, we will investigate certain
properties of these functions.
Using the above equation and the given values for To and T1, the first
several Chebyshev polynomials can be easily calculated; they are
1
T2(x)
2'
3x
T2(x)
= x3
(7.60)
4'
1
T4(x) = x4 – x² +
8'
Proceeding in this way, we can obtain T(x) for any finite integer k. How-
ever, this procedure is very laborious and it would be much better to have
222
Difference Equations
a compact expression giving Tk(x) explicitly in terms of x and k. This can
be easily done because equation (7.59) is a second-order, linear difference
equation, and its general solution can be determined by the techniques of
Chapter 4.
The characteristic equation corresponding to equation (7.59) is
1
= 0.
4
p2
xr +
(7.61)
This equation has the two solutions
(;)
T'1,2 =
(x +
x2
1].
(7.62)
Therefore, the kth Chebyshev polynomial takes the form
G) [A(r1)* + B(r2)*],
(7.63)
2k
where A and B are constants that can be determined by requiring To = 2 and
T1 = x; doing this gives
А+В- 2.
(7.64)
rịA + r2B = 2x,
and
A = B = 1.
(7.65)
Substitution of these values for A and B into equation (7.63) gives an explicit
expression for T; (x):
(Te(2) = () (z + V2 - 1)* + (# – VP - 1)*].
T (x) =
(7.66)
A check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the
results previously given by equation (7.60).
Let us now examine equation (7.66) in more detail. The results of this](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb06891c1-7310-49aa-a044-9ddf722456d6%2F8f47dc35-827d-4202-81f4-3949aa94195e%2Fm566sa_processed.png&w=3840&q=75)
Transcribed Image Text:7.2.3
Chebyshev Polynomials
An Nth-order polynomial in the variable is defined as
PN (x) = .
anaN
+ aN-12N-1
+...+ a1x + ao, aN + 0,
(7.56)
where the a; (for i = 0,1, 2,..., N) are constants, and N is a nonnegative
integer. For example, the following polynomials are, respectively, of zero, first,
and third order:
Po(x) = 3,
P1 (x) = x – 1,
P3(x) = 5x3 + x² – 2x + 4.
(7.57)
A number of special classes of polynomials are of great value for the inves-
tigation of certain problems that arise in both pure and applied mathematics.
In particular, it can be shown that any "reasonable" function of x, defined
over the interval –1 <x < +1, can be written as a sum of the members of any
one of these special classes of polynomials. An example is the class of power
functions whose Nth member is PN = xN. Hence, any (reasonable) function
f (x) has the representation
f(x) = boPo(x) + 61P1 (x) + · · ·+ bN PN (x) + · ..
(7.58)
E bmam.
m=0
The series defined by the second line on the right side of equation (7.58) is
known as a Taylor series. In general, the series expansion of an arbitrary
function contains an unlimited (infinite) number of terms.
An important example of such a class of polynomial functions is the Cheby-
shev polynomials denoted by the symbol T(x). They are defined by the re-
currence formula
() .
Tk-+2 – xT+1 +
Tk = 0,
(7.59)
where |x| < 1, and To = 2, T1 = x. In this section, we will investigate certain
properties of these functions.
Using the above equation and the given values for To and T1, the first
several Chebyshev polynomials can be easily calculated; they are
1
T2(x)
2'
3x
T2(x)
= x3
(7.60)
4'
1
T4(x) = x4 – x² +
8'
Proceeding in this way, we can obtain T(x) for any finite integer k. How-
ever, this procedure is very laborious and it would be much better to have
222
Difference Equations
a compact expression giving Tk(x) explicitly in terms of x and k. This can
be easily done because equation (7.59) is a second-order, linear difference
equation, and its general solution can be determined by the techniques of
Chapter 4.
The characteristic equation corresponding to equation (7.59) is
1
= 0.
4
p2
xr +
(7.61)
This equation has the two solutions
(;)
T'1,2 =
(x +
x2
1].
(7.62)
Therefore, the kth Chebyshev polynomial takes the form
G) [A(r1)* + B(r2)*],
(7.63)
2k
where A and B are constants that can be determined by requiring To = 2 and
T1 = x; doing this gives
А+В- 2.
(7.64)
rịA + r2B = 2x,
and
A = B = 1.
(7.65)
Substitution of these values for A and B into equation (7.63) gives an explicit
expression for T; (x):
(Te(2) = () (z + V2 - 1)* + (# – VP - 1)*].
T (x) =
(7.66)
A check of equation (7.66) for k = 2,3, and 4 shows that it reproduces the
results previously given by equation (7.60).
Let us now examine equation (7.66) in more detail. The results of this
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
