Conjugate Gradient Methods

In the following, A ∈ ℝI x I and b ∈ ℝI are real. We consider a system 9.1.1$$ Ax\, = \,b $$ and assume that 9.1.2$$ A\,is\,positive\,definite. $$ System (1) is associated with the function 9.1.3$$ F\left( x \right): = \,\frac{1}{2}\left\langle {Ax,\,x} \right\rangle \, - \,\left\langle {b,\,x} \right\rangle . $$ The derivative (gradient) of F is $$ F'\left( x \right): = \,\frac{1}{2}\left( {A\, + \,{A^T}} \right)x\, - \,b $$. Since A = AT by assumption (2), the derivative equals 9.1.4$$ F{\text{'}}'\left( x \right)\, = \,grad\,F\left( x \right)\, = \,Ax\, - \,b. $$