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Find a polynomial \(P(x)\) of degree \(\leq 3\) for which $$ \begin{array}{ll} P(0)=y_{1}, & P(1)=y_{2} \\ P^{\prime}(0)=y_{1}^{\prime} & P^{\prime}(1)=y_{2}^{\prime} \end{array} $$ with \(y_{1}, y_{2}, y_{1}^{\prime}, y_{2}^{\prime}\) given constants. The resulting polynomial is called the cubic Hermite interpolating polynomial. Hint: Write $$ P(x)=y_{1} H_{1}(x)+y_{1}^{\prime} H_{2}(x)+y_{2} H_{3}(x)+y_{2}^{\prime} H_{4}(x) $$ with \(H_{1}, H_{2}, H_{3}, H_{4}\) cubic polynomials satisfying appropriate properties, in analogy with (4.15). For example, choose \(H_{1}(x)\) to be a cubic polynomial that satisfies $$ \begin{array}{ll} H_{1}(0)=1, & H_{1}(1)=0 \\ H_{1}^{\prime}(0)=0, & H_{1}^{\prime}(1)=0 \end{array} $$

Step by step solution

01

Define Polynomial Form

Start by expressing the polynomial as follows: \[ P(x) = y_1 H_1(x) + y_1' H_2(x) + y_2 H_3(x) + y_2' H_4(x) \] where \(H_1, H_2, H_3, H_4\) are cubic polynomials to be determined.

02

Define Properties for H鈧�(x)

Find the cubic polynomial \(H_1(x)\) that meets the conditions: \[ H_1(0) = 1, \, H_1(1) = 0, \, H_1'(0) = 0, \, H_1'(1) = 0 \]

03

Calculate H鈧�(x)

By solving these conditions, determine that: \[ H_1(x) = 1 - 3x^2 + 2x^3 \]

04

Define Properties for H鈧�(x)

Find the cubic polynomial \(H_2(x)\) that meets the conditions: \[ H_2(0) = 0, \, H_2(1) = 0, \, H_2'(0) = 1, \, H_2'(1) = 0 \]

05

Calculate H鈧�(x)

By solving these conditions, determine that: \[ H_2(x) = x (1 - 2x + x^2) \]

06

Define Properties for H鈧�(x)

Find the cubic polynomial \(H_3(x)\) that meets the conditions: \[ H_3(0) = 0, \, H_3(1) = 1, \, H_3'(0) = 0, \, H_3'(1) = 0 \]

07

Calculate H鈧�(x)

By solving these conditions, determine that: \[ H_3(x) = 3x^2 - 2x^3 \]

08

Define Properties for H鈧�(x)

Find the cubic polynomial \(H_4(x)\) that meets the conditions: \[ H_4(0) = 0, \, H_4(1) = 0, \, H_4'(0) = 0, \, H_4'(1) = 1 \]

09

Calculate H鈧�(x)

By solving these conditions, determine that: \[ H_4(x) = x^2 (x - 1) \]

10

Write the Final Polynomial

Combine the individual polynomials to get: \[ P(x) = y_1 (1 - 3x^2 + 2x^3) + y_1' (x - 2x^2 + x^3) + y_2 (3x^2 - 2x^3) + y_2' (x^2 (x - 1)) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Interpolation

When dealing with polynomial interpolation, the main goal is to find a polynomial that exactly passes through a given set of points. In other words, we want a polynomial that fits the data points seamlessly. This process is useful in many fields, such as numeric analysis, computer graphics, and data approximation. For instance, in our exercise, Cubic Hermite interpolation is a form of polynomial interpolation involving cubic polynomials which not only match the function values at the given points but also adhere to the provided derivatives.

Cubic Polynomials

Cubic polynomials are polynomials of degree three, which means they have the general form: \[ P(x) = ax^3 + bx^2 + cx + d \] These polynomials are useful because they are flexible enough to model complex shapes but still simple enough to be manageable analytically. In the context of our problem, we define cubic polynomials \(H_1(x), H_2(x), H_3(x)\), and \(H_4(x)\). Each polynomial is carefully constructed to meet specific conditions. For example, \(H_1(x)\) must satisfy the properties: \(H_1(0) = 1, H_1(1) = 0, H_1'(0) = 0, H_1'(1) = 0\). These constraints ensure that the cubic Hermite polynomial is formed correctly.

Derivatives in Polynomial Interpolation

Derivatives play a crucial role in polynomial interpolation. They ensure that the polynomial not only matches the function values but also has the correct slopes (derivatives) at the key points. In the exercise, \(P(x)\) is a cubic Hermite interpolating polynomial which matches both the function values (\(y_1, y_2\)) and their first derivatives (\(y_1', y_2'\)) at specific points. To achieve this, we solve for cubic polynomials like \(H_2(x)\), which meets the derivative condition \(H_2'(0) = 1\), thus ensuring the correct slope at that point. This helps in creating a smooth and accurate approximation for the entire interval.

Numerical Analysis

Numerical analysis involves mathematical methods to solve numeric problems. In polynomial interpolation, numerical analysis helps us develop algorithms to find polynomials fitting a set of data points. For instance, Cubic Hermite interpolation is a numeric method used to construct a polynomial that not only fits the data points but also maintains proper derivative values at those points. This method is particularly useful in applications requiring smooth curves through given data, such as in computer graphics and data modeling. By carefully applying these numerical methods, we can achieve very precise and accurate results, as demonstrated by our step-by-step solution above.