In this problem, we discuss one way that computers can handle polynomials in their arith- metic. This is an important application for all sorts of software, especially splines in computer graphics. Machine learning specialists also use representations like these to find "best fit" curves. Consider the set Pa(x) of polynomials with degree at most d and variable x. Every polynomial in this set can be written as p(x) = ao + a1x² + a2x² + · .. %3D Consider the polynomial p(x) = -4+ 2x + 6x². Use matrix multiplication to а. find a vector ao a = a2 So that ao p(x) = [1 x x²] ai a2 b. We can use this technique to find the coefficients of a polynomial, so long as we know the outputs of that polynomial. Suppose that q(x) = bo + b1x + b2x². If we know that -1, q(xo) -11 0, q(x1) -7 X2 1, q(x2) -9 Then we can write the polynomial values as a system of linear equations. [1 xo (xo)²] bo 1 x1 (x1)²| |b1 x2 (x2)²] [b2 [q(x0)] q(x1) La(x2)] Fill in the numbers given above and use Gaussian Elimination to find the coefficients bo, b1, b2. State which row operations you use at each step. (Note: This method is actually used in some computer systems to find a curve passing through many points. Think data fitting, graphics calculation, etc.)
In this problem, we discuss one way that computers can handle polynomials in their arith- metic. This is an important application for all sorts of software, especially splines in computer graphics. Machine learning specialists also use representations like these to find "best fit" curves. Consider the set Pa(x) of polynomials with degree at most d and variable x. Every polynomial in this set can be written as p(x) = ao + a1x² + a2x² + · .. %3D Consider the polynomial p(x) = -4+ 2x + 6x². Use matrix multiplication to а. find a vector ao a = a2 So that ao p(x) = [1 x x²] ai a2 b. We can use this technique to find the coefficients of a polynomial, so long as we know the outputs of that polynomial. Suppose that q(x) = bo + b1x + b2x². If we know that -1, q(xo) -11 0, q(x1) -7 X2 1, q(x2) -9 Then we can write the polynomial values as a system of linear equations. [1 xo (xo)²] bo 1 x1 (x1)²| |b1 x2 (x2)²] [b2 [q(x0)] q(x1) La(x2)] Fill in the numbers given above and use Gaussian Elimination to find the coefficients bo, b1, b2. State which row operations you use at each step. (Note: This method is actually used in some computer systems to find a curve passing through many points. Think data fitting, graphics calculation, etc.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![In this problem, we discuss one way that computers can handle polynomials in their arith-
metic. This is an important application for all sorts of software, especially splines in computer
graphics. Machine learning specialists also use representations like these to find "best fit"
curves. Consider the set Pa(x) of polynomials with degree at most d and variable x. Every
polynomial in this set can be written as
p(x)
= ao + a1x + a2x? +
+ agrd.
Consider the polynomial p(x) = -4+ 2x + 6x². Use matrix multiplication to
а.
find a vector
ao
а —
A2
So that
ao
p(x) = [1 x x²] a1
A2
b.
We can use this technique to find the coefficients of a polynomial, so long as
we know the outputs of that polynomial. Suppose that q(x) = bo + b1x + b2x². If we
know that
-1,
q(xo)
= (Ox)b
0,
q(x1)
q(x2)
1,
-11
X1
-7
X2
-9
Then we can write the polynomial values as a system of linear equations.
1 xo (xo)2] bo
1 x1 (21)?
1 x2 (x2)²] [b2
[q(xo)]
q(x1)
Lq(x2).
Fill in the numbers given above and use Gaussian Elimination to find the coefficients
bo, b1, b2. State which row operations you use at each step. (Note: This method is
actually used in some computer systems to find a curve passing through many points.
Think data fitting, graphics calculation, etc.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6431b385-68cb-483f-9219-6877a56d56cf%2F4103c350-716a-4538-970d-95b24ad9f4c3%2Fr6umsgp_processed.png&w=3840&q=75)
Transcribed Image Text:In this problem, we discuss one way that computers can handle polynomials in their arith-
metic. This is an important application for all sorts of software, especially splines in computer
graphics. Machine learning specialists also use representations like these to find "best fit"
curves. Consider the set Pa(x) of polynomials with degree at most d and variable x. Every
polynomial in this set can be written as
p(x)
= ao + a1x + a2x? +
+ agrd.
Consider the polynomial p(x) = -4+ 2x + 6x². Use matrix multiplication to
а.
find a vector
ao
а —
A2
So that
ao
p(x) = [1 x x²] a1
A2
b.
We can use this technique to find the coefficients of a polynomial, so long as
we know the outputs of that polynomial. Suppose that q(x) = bo + b1x + b2x². If we
know that
-1,
q(xo)
= (Ox)b
0,
q(x1)
q(x2)
1,
-11
X1
-7
X2
-9
Then we can write the polynomial values as a system of linear equations.
1 xo (xo)2] bo
1 x1 (21)?
1 x2 (x2)²] [b2
[q(xo)]
q(x1)
Lq(x2).
Fill in the numbers given above and use Gaussian Elimination to find the coefficients
bo, b1, b2. State which row operations you use at each step. (Note: This method is
actually used in some computer systems to find a curve passing through many points.
Think data fitting, graphics calculation, etc.)
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