年>$24万 달성; {2037}~2064 年,총$6万 – Roth, SEP IRA / Solo 401(k)；JINU "$1B" (재)도전 生活
2024~2028
毎日4時間の学習 @McDonald’s
USA.CsamiGroup.com
Foothill (AB 540)
CPA.CsamiGroup.com
2028~2037
JINU 毎9年 1번 도전
Mathematics MA, IndState
Statistics MS, UAF (stipend)
毎日4時間の学習 @McDonald’s 生活 ~2037
“봄 나들이-PhD”
2055(91歳)~
~2037 (73歳)
2037~
2055 (91歳)~
2024
2025
..
2027
2028
..
2031
..
2034
..
2037 “毎日4時間の学習 生活”
１日、$0.62 >> 총, ~$1,100 (1/2)
~2028 웹 55개 + SSDI EBT Cash Aid
REMOTE QBO
USA.CsamiGroup.com
CPA.CsamiGroup.com
JINU “미주장학재단” version
Newton's method, also known as the Newton-Raphson method, is an iterative numerical technique used to find roots of real-valued functions. The reason it converges to a root can be explained through several key concepts:
### 1. **Function and Derivative**:
Newton's method utilizes both the function $latex f(x)$ and its derivative $latex f'(x)$ to estimate the root. The formula for the iteration is given by:
$latex x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$
This formula gives the next approximation $latex x_{n+1}$ based on the current approximation $latex x_n$.
### 2. **Tangent Line Approximation**:
The essence of Newton's method lies in the geometric interpretation. At each iteration:
- A tangent line is drawn to the function at the point $latex (x_n, f(x_n))$.
- The x-intercept of this tangent line is calculated as the next approximation $latex x_{n+1}$.
The tangent line approximates the function locally, which means if the initial guess $latex x_n$ is close to the actual root, the next guess $latex x_{n+1}$ will typically be closer to the root.
### 3. **Convergence**:
Newton's method exhibits quadratic convergence near the root under certain conditions:
- If $latex x_n$ is sufficiently close to the root $latex r$, then $latex f(x_n)$ will be small, and $latex f'(x_n)$ will be significant.
- The approximation becomes increasingly accurate, and the error decreases rapidly with each iteration.
### 4. **Conditions for Convergence**:
For Newton's method to converge to a root:
- The function $latex f$ must be differentiable in the neighborhood of the root.
- The derivative $latex f'(x)$ should not be zero at the root; otherwise, the method fails as the formula will involve division by zero.
- The initial guess $latex x_0$ should be sufficiently close to the actual root.
### 5. **Mathematical Justification**:
Mathematically, if we denote the root as $latex r$ and assume $latex f(r) = 0$, we can analyze the Taylor series expansion around the root:
$latex f(x_n) = f(r) + f'(r)(x_n - r) + \frac{f''(r)}{2}(x_n - r)^2 + \ldots$
Since $latex f(r) = 0$, this simplifies to:
$latex f(x_n) \approx f'(r)(x_n - r) + \frac{f''(r)}{2}(x_n - r)^2$
The behavior of this expansion shows that as $latex x_n$ approaches $latex r$, the next iteration $latex x_{n+1}$ rapidly approaches the root.
### Conclusion
In summary, Newton's method converges to a root because it uses local linear approximations (tangent lines) to refine guesses iteratively. Given suitable conditions, this method can yield highly accurate approximations to the root with relatively few iterations.
毎日4時間の学習 @McDonald’s 生活
USA.CsamiGroup.com
Foothill
BBA, EA QBO CPA
年$78,000 달성 後、毎年、月+$1,000
~2028~2031~2034~2037 은퇴 I.II.III.
JINU 毎9年 × 1번 도전
2037~2064 은퇴 연금 적립
(6)(6)(6) ~2055
JINU 毎9年 × 2번 도전
(9)(9)(9)(9)(9) ~2100
JINU 毎9年 × 5번 도전