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162 chapter 8We now have a solvable set of nine linear equations in the nine unknowns ∆x i ,which we express as a single matrix equationf 1 + ∂f 1 /∂x 1 ∆x 1 + ∂f 1 /∂x 2 ∆x 2 + ···+ ∂f 1 /∂x 9 ∆x 9 =0,f 2 + ∂f 2 /∂x 1 ∆x 1 + ∂f 2 /∂x 2 ∆x 2 + ···+ ∂f 2 /∂x 9 ∆x 9 =0,. ..f 9 + ∂f 9 /∂x 1 ∆x 1 + ∂f 9 /∂x 2 ∆x 2 + ···+ ∂f 9 /∂x 9 ∆x 9 =0,⎛ ⎞ ⎛⎞ ⎛ ⎞f 1∂f 1 /∂x 1 ∂f 1 /∂x 2 ··· ∂f 1 /∂x 9 ∆x 1f 2∂f 2 /∂x 1 ∂f 2 /∂x 2 ··· ∂f 2 /∂x 9∆x 2⎜. +.. ⎟ ⎜ .⎝ ⎠ ⎝ ..⎟ ⎜ . =0. (8.15).. ⎟⎠ ⎝ ⎠f 9 ∂f 9 /∂x 1 ∂f 9 /∂x 2 ··· ∂f 9 /∂x 9∆x 9Note now that the derivatives and the f’s are all evaluated at known values of thex i ’s, so only the vector of the ∆x i values is unknown. We write this equation inmatrix notation asf + F ′ ∆x =0, ⇒ F ′ ∆x = −f, (8.16)⎛ ⎞ ⎛ ⎞ ⎛⎞∆x 1f 1∂f 1 /∂x 1 ··· ∂f 1 /∂x 9∆x 2f 2∂f ∆x =⎜ . , f =.. ⎟ ⎜. , F ′ 2 /∂x 1 ··· ∂f 2 /∂x 9=... ⎟ ⎜ .⎝ ⎠ ⎝ ⎠ ⎝ ..⎟⎠∆x 9 f 9 ∂f 9 /∂x 1 ··· ∂f 9 /∂x 9Here we use bold to emphasize the vector nature of the columns of f i and ∆x i valuesand call the matrix of the derivatives F ′ (it is also sometimes called J because it isthe Jacobian matrix).The equation F ′ ∆x = −f is in the standard form for the solution of a linearequation (often written Ax = b), where ∆x is the vector of unknowns and b = −f.Matrix equations are solved using the techniques of linear algebra, and in thesections to follow we shall show how to do that on a computer. In a formal (andsometimes practical) sense, the solution of (8.16) is obtained by multiplying bothsides of the equation by the inverse of the F ′ matrix:∆x = −F ′ −1 f, (8.17)where the inverse must exist if there is to be a unique solution. Although we aredealing with matrices now, this solution is identical in form to that of the 1-Dproblem, ∆x = −(1/f ′ )f. In fact, one of the reasons we use formal or abstractnotation for matrices is to reveal the simplicity that lies within.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 162