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given is this function $y = N \cdot e^st$.

I have to transform it to a "linear"(least square approx.)

$\implies y = \ln(N)+ st$

How do I put this to a Matrix form $A \cdot x = b$

$N$ and $s$ are unknown.

The only problem that I have is I don't know how to represent $\ln(N)$ in the Matrix because $N$ is unknown.

I know that the Matrix should look like this: \begin{bmatrix}" "&t\\\end{bmatrix} multiple by the unknowns \begin{bmatrix}N\\s\end{bmatrix}

What should I write in "" so when I multiply the matrix, it gives the $\ln(N)$

NoobProgrammer
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1 Answers1

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So if we have the formula $$y_i=Ne^{st_i},$$ where $N,s$ are variables, and $y_i,t_i$ are known, we can take the log on both sides leading to: $$ \text{ln}(y_i) = \text{ln(N)}+st_i$$ Now supposing, we have $M$ such equations, we can stack them into a matrix-vector equation, $$ \begin{bmatrix} \text{ln}(y_1) \\ \vdots \\ \text{ln}(y_M) \end{bmatrix} = \begin{bmatrix} 1 & t_1 \\ \vdots & \vdots \\ 1 & t_M \end{bmatrix} \begin{bmatrix} \text{ln}(N) \\ s \end{bmatrix}$$ In which we recognize, $b=Ax$. By solving the above (assuming we have 2 or more independent equations) we will obtain $s$ and $\text{ln}(N)$, which will in turn lead us to $N$.

wiegert
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