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I have been trying to solve a particular problem that establishes both the existence and uniqueness of a cubic hermite interpolant on some generic interval $[a,b]$. Briefly, for a function $f$ we require a cubic polynomial $p$ that satisfies

$$p(a) = f(a), \ p^{\prime}(a) = f^{\prime}(a), $$ $$p(b) = f(b), \ p^{\prime}(b) = f^{\prime}(b). $$

I am aware that there are several approaches for this but the approach that my book recommends is to use the fact (which I have successfully proven) that the differential equation $u^{(4)}(x) = f(x)$ with $u(a) = u(b) = u^{\prime}(a) = u^{\prime}(b) = 0$ has a unique solution. Using this one can then easily show that the cubic hermite interpolant is unique but I am baffled as to how this proves existence and therefore I would appreciate some help.

I should mention that this is an exercise from Prenter's "Splines and Variational Methods".

Thank you in advance.

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The problem of existence and uniqueness of a cubic polynomial passing through two given points with two given slopes reduces to a linear system of four equations in four unknowns, as follows. Take the most general cubic, $P(x)=\alpha x^3+\beta x^2 + \gamma x + \delta$. Then $P'(x)=3 \alpha x^2 + 2\beta x + \gamma$, and you want: $$ \begin{aligned} \alpha a^3 + \beta a^2 + \gamma a + \delta &= f(a)\\ \alpha b^3 + \beta b^2 + \gamma b + \delta &= f(b)\\ 3\alpha a^2 + 2\beta a + \gamma &= f'(a)\\ 2\alpha b^2 + 2\beta b + \gamma &= f'(b)\\ \end{aligned} $$ The corresponding matrix of coefficients is $$ A= \begin{pmatrix} a^3 & a^2 & a & 1\\ b^3 & b^2 & b & 1\\ 3a^2 & 2a & 1 & 0 \\ 3b^2 & 2b & 1 & 0 \end{pmatrix} $$

A straighforward calculation shows that the determinant of this matrix is $-(a-b)^4$, which is nonzero if we assume that $a\neq b$, and of course we assume that. From Linear Algebra we know that the above linear system of equations will have a unique solution. That is, there exists a solution, and it is unique. Moreover, using standard techniques it is possible to express the coefficients of the interpolating polynomial as functions of the input data $(f(a),f(b),f'(a),f'(b))$.

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  • $\begingroup$ Yes I know all this but my question is about the specific exercise. How does it prove existence, as Prenter claims? $\endgroup$ – JohnK Apr 11 '19 at 10:24
  • $\begingroup$ I don't see how existence of an interpolating cubic follows from the uniqueness of the solution to the IVP $u^{(4)}(x)=f(x)$. The way things are usually done, is that if you know that this differential equation has a unique solution you look at $f-P$ where you already have proven that $P$ exists. Maybe you are just reading too much into the exercise. $\endgroup$ – uniquesolution Apr 11 '19 at 10:41
  • $\begingroup$ Perhaps you are right, thanks for your contribution. $\endgroup$ – JohnK Apr 11 '19 at 11:13

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