How to determine the truncation error with products and quotients

If I have an equation given by

$$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$

and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is truncated at the $B^{th}$ order, $c$ is truncated at the $C^{th}$ order, and $d$ is truncated at the $D^{th}$ order, how do I go about determining the order of the truncation error for the full equation $Y$?

Given two truncated Laurent series, $f = p(x)+O(x^p)$, $g = q(x)+O(x^q)$, the orders will be $$f+g = p(x) + q(x) + O(x^{\min(p,q)}),$$ $$fg = p(x) q(x) + O(x^{\min(p+q_0, q+p_0)}),$$ $$1/g = 1/q(x) + O(x^{q-2q_0}).$$ where $p_0$ and $q_0$ are the degrees (possibly negative) of the first nonzero monomials in $p$ and $q$. So for each subexpression it is necessary to determine the leading term (which is possibly non-constant, and might be singular), then it is straightforward algebra.

• I follow you with the addition and multiplication, but I'm a bit confused on 1/g. Where does the 2 come from? Also...how does raising a term to a power influence the truncation error? Following your example, if it's $f^2$ does the truncation error become $O(x^{2p})$ or $O(x^{2+p})$? Could you possibly cater your answer to directly reflect the equation I presented? Given 4 truncated Laurent series... Commented Jul 7, 2016 at 23:21
• You've left me hanging...just thought I'd try to get your attention to see if you're still available to offer further insight. Commented Jul 19, 2016 at 23:00
• could you possibly elaborate more based on my comment above? Commented Aug 4, 2016 at 20:05

It uses an approach similar to total differentiation:
S = P * G ==> error : D(S) = D(P) * G + P * D(G) ,
since O(h^m) , "h" is scaled such that h < 1
The leading error is the one with lower "m" power

S = x^m => error : DS = m * X^(m-1) * DX

So on ...

• Welcome to Computational Science. I am not sure how this answers the original question, especially with the used notation. Could you edit it to be more reflective of the original question? Also, we have support for MathJax which allows proper typing of formulas. Commented Jan 7, 2023 at 0:18
The text passage contains a mathematical question about determining the order of truncation error for a given equation when the variables are expanded in Taylor series and truncated at different orders. The passage also includes a partial answer that provides some rules for determining the order of truncation error for addition, multiplication, and division of truncated series, but it does not fully resolve the original question.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
How to determine the truncation error with products and quotients

If I have an equation given by

$$\displaystyle Y = \frac{a^2}{d^2}\frac{(1-c^2\frac{c}{a})}{(1-b^2)}$$

and I expand $a,b,c,d$ in a Taylor series, where $a$ is truncated at the $A^{th}$ order, $b$ is truncated at the $B^{th}$ order, $c$ is truncated at the $C^{th}$ order, and $d$ is truncated at the $D^{th}$ order, how do I go about determining the order of the truncation error for the full equation $Y$?

Given two truncated Laurent series, $f = p(x)+O(x^p)$, $g = q(x)+O(x^q)$, the orders will be $$f+g = p(x) + q(x) + O(x^{\min(p,q)}),$$ $$fg = p(x) q(x) + O(x^{\min(p+q_0, q+p_0)}),$$ $$1/g = 1/q(x) + O(x^{q-2q_0}).$$ where $p_0$ and $q_0$ are the degrees (possibly negative) of the first nonzero monomials in $p$ and $q$. So for each subexpression it is necessary to determine the leading term (which is possibly non-constant, and might be singular), then it is straightforward algebra.

• I follow you with the addition and multiplication, but I'm a bit confused on 1/g. Where does the 2 come from? Also...how does raising a term to a power influence the truncation error? Following your example, if it's $f^2$ does the truncation error become $O(x^{2p})$ or $O(x^{2+p})$? Could you possibly cater your answer to directly reflect the equation I presented? Given 4 truncated Laurent series... Commented Jul 7, 2016 at 23:21
• You've left me hanging...just thought I'd try to get your attention to see if you're still available to offer further insight. Commented Jul 19, 2016 at 23:00
• could you possibly elaborate more based on my comment above? Commented Aug 4, 2016 at 20:05

It uses an approach similar to total differentiation:
S = P * G ==> error : D(S) = D(P) * G + P * D(G) ,
since O(h^m) , "h" is scaled such that h < 1
The leading error is the one with lower "m" power

S = x^m => error : DS = m * X^(m-1) * DX

So on ...

• Welcome to Computational Science. I am not sure how this answers the original question, especially with the used notation. Could you edit it to be more reflective of the original question? Also, we have support for MathJax which allows proper typing of formulas. Commented Jan 7, 2023 at 0:18
user
The text passage contains a mathematical question about determining the order of truncation error for a given equation when the variables are expanded in Taylor series and truncated at different orders. The passage also includes a partial answer that provides some rules for determining the order of truncation error for addition, multiplication, and division of truncated series, but it does not fully resolve the original question.

TYPE: SINGLE_QUESTION_WITH_ANSWER