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Section 11.3 Details of Broyden's method

Recall from Section 11.1 that we seek to improve our guess for the inverse of Jacobian by finding a matrix B1 such that

B1(F(x1)−F(x0))=x1−x0

To simplify notation, let h=x1−x0 and w=F(x1)−F(x0); both these vectors are known. We need

B1w=h

We already know how to find a matrix that satisfies (11.3.1), by taking an outer product multiplied by a scalar (11.2.2). But this would be a bad, illogical choice for \(B_1\text{.}\) Why?

We want B1 to have some similarity to B0 in the hope that the process of improving guesses B0,B1,… will converge to something, rather than just jump around. We want to improve the previous guess, not replace it entirely. So, let B1=(I+M)B0=B0+MB0 where M can be constructed from an outer product as in (11.2.2). Namely, we want

B0w+MB0w=h

so

MB0w=h−B0w

According to (11.2.2) we can achieve this by letting

M=1hTB0w(h−B0w)hT

where the chose the term b to be h. To summarize, the formula for updating our guess for inverse Jacobian is

B1=B0+1hTB0w(h−B0w)hTB0

It looks messy, but at least the derivation was logical.

The matrix B1 is still a guess, it may be very different from the actual inverse of Jacobian matrix. But it is a more educated guess, and as this process repeats, these repeated corrections will produce more accurate guesses.