I have the question and the answer for this problem I just want to understand the concepts and logic of how to do it step by step LUsing Taylor's thearem, we can obtain a formula to calculate he sine of Ssin(x) - x- + (-1)(2n - 1)!Write a function detinition to calculate the sine of x using the formula above. Your function shouldteminate when the absolute value of the last term calculated is less than 1.0 x 10. You may not use anyfunctions supplied in the cmath.h header file (ie. you cannor use the powx.y) function Your sinefunction may call other functions, and in this case, you must also wrie the definitions of those functions.You do nor need to comment anything, although, a well-placed comment may provide clarity for thegrader.Bonus (2 pes): Briefly explain why the sine function above will always converge. You will only get thebonus points if the sine function is wrimen.Single Function VersionMultiple Function Versiondouble sin(double x) (int i;int n- 1int sign;double eps - 1.De-4;double sum- 0.0;double pow;lang fact;double this term;do {pow - 1.0;Pact- 1for (1-1; ie-2*n-1; +1) {double sint double x) (int a- 1;double eps - 1.0e-4;double sum - 0.0;double this teradothis tern-sign(n-1)"pow(x, 2"a-1)/fact(2'n-1):sum this tern;ir (this tern <0.0) this tern --this term;*n;} while (this tern eps);return sum;pow x;fact iir ((n-aa - 1)sign --1;elsesign - 1;this tern- sign'pow/fact;int sign(int n) {if (ns2 -- 1)return -1return 1;sum - this_term;if (this_ter < 0.0)this tern --this termdouble powdouble K, int a) {int Adouble p- 1.0for (i-0; ien; **1)} while (this tera eps):Feturn sim;P- x;return p:lang fact (int n) {int i;long f-1for (1-1; ien; 1) {return f

The code for the given function is given below.

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for this problem I ju

C++ Programming: From Problem Analysis to Program Design

8th Edition

ISBN:9781337102087

Author:D. S. Malik

Publisher:D. S. Malik

Chapter15: Recursion

Section: Chapter Questions

Problem 20PE

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Concept explainers

LUP Decomposition

LU decomposition stands for Lower-Upper decomposition. LU decomposition is the factorization of the lower triangular matrix and an upper triangular matrix. It is very useful in linear algebra and numerical analysis. A permutation matrix is usually included in the product. Gaussian elimination in a matrix form is known as LU decomposition, When inverting a matrix or evaluating its determinant, this is a crucial step. Tadeusz Banachiewicz, a Polish mathematician, first proposed the LU decomposition in 1938.

Question

I have the question and the answer for this problem I just want to understand the concepts and logic of how to do it step by step

LUsing Taylor's thearem, we can obtain a formula to calculate he sine of S
sin(x) - x
- + (-1)
(2n - 1)!
Write a function detinition to calculate the sine of x using the formula above. Your function should
teminate when the absolute value of the last term calculated is less than 1.0 x 10. You may not use any
functions supplied in the cmath.h header file (ie. you cannor use the powx.y) function Your sine
function may call other functions, and in this case, you must also wrie the definitions of those functions.
You do nor need to comment anything, although, a well-placed comment may provide clarity for the
grader.
Bonus (2 pes): Briefly explain why the sine function above will always converge. You will only get the
bonus points if the sine function is wrimen.
Single Function Version
Multiple Function Version
double sin(double x) (
int i;
int n- 1
int sign;
double eps - 1.De-4;
double sum- 0.0;
double pow;
lang fact;
double this term;
do {
pow - 1.0;
Pact- 1
for (1-1; ie-2*n-1; +1) {
double sint double x) (
int a- 1;
double eps - 1.0e-4;
double sum - 0.0;
double this tera
do
this tern-sign(n-1)"pow(x, 2"a-
1)/fact(2'n-1):
sum this tern;
ir (this tern <0.0) this tern -
-this term;
*n;
} while (this tern eps);
return sum;
pow x;
fact i
ir ((n-aa - 1)
sign --1;
else
sign - 1;
this tern- sign'pow/fact;
int sign(int n) {
if (ns2 -- 1)
return -1
return 1;
sum - this_term;
if (this_ter < 0.0)
this tern --this term
double powdouble K, int a) {
int A
double p- 1.0
for (i-0; ien; **1)
} while (this tera eps):
Feturn sim;
P- x;
return p:
lang fact (int n) {
int i;
long f-1
for (1-1; ien; 1) {
return f

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