Residual (numerical analysis)

Loosely speaking, a residual is the error in a result.^[1] To be precise, suppose we want to find x such that

f(x)=b.

Given an approximation x₀ of x, the residual is

b-f(x_{0})

that is, "what is left of the right hand side" after subtracting f(x₀)" (thus, the name "residual": what is left, the rest). On the other hand, the error is

x-x_{0}

If the exact value of x is not known, the residual can be computed, whereas the error cannot.

^ Shewchuk, Jonathan Richard (1994). "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" (PDF). p. 6.

[1] Shewchuk, Jonathan Richard (1994). "An Introduction to the Conjugate Gradient Method Without the Agonizing Pain" (PDF). p. 6.

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