I Residual of an algebraic equation

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To solve the equations f(x,y)=0 and g(x,y)=0, Newton iteration is employed, allowing for the definition of a residual at each iteration. The term 'residual of x' or 'residual of y' refers to the individual components of the overall residual from the iteration process. This suggests that the residual should be analyzed separately for each variable rather than as a collective measure of the equations. Understanding these residuals is crucial for assessing convergence in the iterative process. Clarifying these terms enhances the comprehension of the Newton iteration method in solving algebraic equations.
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To solve a set of equations f(x,y)=0, g(x,y)=0, where x, y, f, g are scalars, use Newton iteration. At each iteration step i, can certainly define the residual of this set of equations. But what's meant by the term 'residual of x' or 'residual of y'? Not the 'residual of the equations'?
 
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My guess would be take the residual of the iteration and look at the components separately.
 

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