... easiest¹

However, no derivation that I have encountered is very straightforward. The approach I take is found in Dennis and Schnabel [1], an excellent reference for secant methods.

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... Algebra²

The Fundamental Theorem of Linear Algebra states, in part, that the rank of any matrix plus the dimension of its null space must equal the number of columns.

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... it.³

I will explain how to compute finite-difference estimates of the Jacobian later.

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... equation.⁴

The proof is left as an exercise.

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... update:⁵

The formula is due to Broyden, Fletcher, Goldfarb, and Shanno.

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