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![Civen t .(+10+3+1) modu h(A)-並76 h(3): ±1463 ะ (с5), 10)cof+ 3(5)+7).rnod l] = (125+456 + 15+7)modll ht5) h(s) h(5) s 4341 h](http://img.homeworklib.com/images/0d0c4f8c-887b-4a09-9e35-8842fd66254b.png?x-oss-process=image/resize,w_560)
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1. Consider Shamir's Lagrange interpolating polynomial key threshold scheme. Let t p-11, K-Tand C...
X 1 1 11 (C). The polynomial p() = 4 - 2 + 2? - +1...
X 1 1 11 (C). The polynomial p() = 4 - 2 + 2? - +1 has the values shown. -2-1 01 | 2 3 p(x) 31 5 61 Find a polynomial (2) that takes these values (you don't need expand it): -2 -1 0 1 2 3 9(x) 31 5 11 30 (Hint: This can be done with little work. Try the Lagrange form.) 1 1
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
Consider the following set of data x f(x) 3 6 4 3 5 8 1. Use...
Consider the following set of data x f(x) 3 6 4 3 5 8 1. Use and order Newton polynomial to find f (4.5). 2. Use and order Lagrange polynomial to find f (4.5). You should get the same answer using both methods they are just different representations of a quadratic (i.e., 2nd order) interpolating polynomial.
Consider the function f(x) := v/x= x1/2. 6. (a) Give the Taylor polynomial P(x) of degree 5 about a1 of this function (b) Give the nested representation of the polynomial Qs()Ps((t)) where t -1 (...
Consider the function f(x) := v/x= x1/2. 6. (a) Give the Taylor polynomial P(x) of degree 5 about a1 of this function (b) Give the nested representation of the polynomial Qs()Ps((t)) where t -1 ((t)+1). (c) Using the nested multiplication method (also called Horner's algorithm), compute the approximation Ps (1.2) to V (give at least 12 significant digits of P(1.2)) (d) Without using the exact value of 12, compute by hand an upper bound on the absolute error V1.2 A(1.21...
2. (a) Let P, =Span{1, x, x?, x°, x*} be the collection of polynomial with degree...
2. (a) Let P, =Span{1, x, x?, x°, x*} be the collection of polynomial with degree at most 4. Con- sider subspace H = Span{1,x, x*}. Prove that H is a subspace of Pg. Find a basis for the subspace H. (b) Now consider the differential operator D : H P, defined by D(1) = 0, D(x) = 1 and D(x3)=3x2. Why this defined linear operator D:H-P? Is the map Donto? Is the map Done-to-one?
2. Consider the polynomials 0-k (z) := (1 + z) for k-0,..., 10 and let B-bo,b1bo) can be shown th...
2. Consider the polynomials 0-k (z) := (1 + z) for k-0,..., 10 and let B-bo,b1bo) can be shown that B is a basis for Pio the vector space of polynomials of degree at most 10. (You do not need to prove this.) Let Pk (z)-rk for k = 0, 1, . . . , 10, so that S = {po, pi, . . . , pio) is the standard basis for P10. Use Mathematica to: (a) Compute the change...
2. Consider the polynomial p = x3 + x +4 € Z5 [2]. Let q =...
2. Consider the polynomial p = x3 + x +4 € Z5 [2]. Let q = 3x +2 € Z5 [2]. (a) Is p reducible or irreducible? Prove your claim. (b) Are there any degree 2 polynomials in [g],? Explain. (c) List all degree 3 polynomials in [g]p. (d) (ungraded for thought) How many degree 4 polynomials are in (q),? Degree 5?
matlab The error function is a mathematical function that frequently arises in probability and statistics. It also c...
matlab
The error function is a mathematical function that frequently arises in probability and statistics. It also can show up in the solution to some partial differential equations, particularly those arising in heat and mass transfer applications. The error function is defined as 2 e-t dt picture attached This function is actually built-in to MATLAB as the command erf, and here we'll use that function to compute a "true value" with which we can compare results of two interpolation approaches....
2.11. Let x(t) 11(1-3)-u(t-5) and h(t) = e-3t11(1). (a) Compute y(i) - x(t) * h(t)
2.11. Let x(t) 11(1-3)-u(t-5) and h(t) = e-3t11(1). (a) Compute y(i) - x(t) * h(t)
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 +...
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
X 1 1 11 (C). The polynomial p() = 4 - 2 + 2? - +1 has the values shown. -2-1 01 | 2 3 p(x) 31 5 61 Find a polynomial (2) that takes these values (you don't need expand it): -2 -1 0 1 2 3 9(x) 31 5 11 30 (Hint: This can be done with little work. Try the Lagrange form.) 1 1
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
Consider the following set of data x f(x) 3 6 4 3 5 8 1. Use and order Newton polynomial to find f (4.5). 2. Use and order Lagrange polynomial to find f (4.5). You should get the same answer using both methods they are just different representations of a quadratic (i.e., 2nd order) interpolating polynomial.
Consider the function f(x) := v/x= x1/2. 6. (a) Give the Taylor polynomial P(x) of degree 5 about a1 of this function (b) Give the nested representation of the polynomial Qs()Ps((t)) where t -1 ((t)+1). (c) Using the nested multiplication method (also called Horner's algorithm), compute the approximation Ps (1.2) to V (give at least 12 significant digits of P(1.2)) (d) Without using the exact value of 12, compute by hand an upper bound on the absolute error V1.2 A(1.21...
2. (a) Let P, =Span{1, x, x?, x°, x*} be the collection of polynomial with degree at most 4. Con- sider subspace H = Span{1,x, x*}. Prove that H is a subspace of Pg. Find a basis for the subspace H. (b) Now consider the differential operator D : H P, defined by D(1) = 0, D(x) = 1 and D(x3)=3x2. Why this defined linear operator D:H-P? Is the map Donto? Is the map Done-to-one?
2. Consider the polynomials 0-k (z) := (1 + z) for k-0,..., 10 and let B-bo,b1bo) can be shown that B is a basis for Pio the vector space of polynomials of degree at most 10. (You do not need to prove this.) Let Pk (z)-rk for k = 0, 1, . . . , 10, so that S = {po, pi, . . . , pio) is the standard basis for P10. Use Mathematica to: (a) Compute the change...
2. Consider the polynomial p = x3 + x +4 € Z5 [2]. Let q = 3x +2 € Z5 [2]. (a) Is p reducible or irreducible? Prove your claim. (b) Are there any degree 2 polynomials in [g],? Explain. (c) List all degree 3 polynomials in [g]p. (d) (ungraded for thought) How many degree 4 polynomials are in (q),? Degree 5?
matlab
The error function is a mathematical function that frequently arises in probability and statistics. It also can show up in the solution to some partial differential equations, particularly those arising in heat and mass transfer applications. The error function is defined as 2 e-t dt picture attached This function is actually built-in to MATLAB as the command erf, and here we'll use that function to compute a "true value" with which we can compare results of two interpolation approaches....
2.11. Let x(t) 11(1-3)-u(t-5) and h(t) = e-3t11(1). (a) Compute y(i) - x(t) * h(t)
Problem 4. (i) Let R> 2/14Z and consider the polynomial ring R[d]. Let A(z) 4 + 2r3 + 3r2 + 4x + 5 and B(x) 37 be elements of R]. Find q(x) and r(x) in R] such that: A(x)-q(z)E(z) + r(z) and deg(r) < 2. (2pts) (ii) Let R- Z/11Z, write down the table of squares in R as follows. For every a E R (there are 11 such elements), find a2. Here you are required to express the final...
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