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?? ??? ?? ?? ???(???)? ??? ?? ?? ?? ????? ????? ??????(???)? ??. ??? ??? ?????? ????, ??????? ? ?? ??? ??? n? ??? ??? n???? ????. (???, n=2)

百度 回顾这些提案，有哪些悄然推动了政策改革和社会变化，改变着你我的日常生活？推动完善事中事后监管体制：叫百姓少受骗、好办事2016年，致公党中央在全国政协十二届四次会议上提出了《关于加快构建和简政放权相对应的事中事后监管体制机制的建议》。

?????? ?? ??????(??: Conjugate Gradient Method, ???: 共役勾配法)?? ???? ??? ?? ??????(陽-準定符號行列, ??: positive-semidefinite matrix)? ?? ???? ?? ??? ?? ??????. ?? ?? ???????? ???, ??? ???? ??? ?? ???? ?? ??? ?? ? ?? ?? ???? ?? ???? ????. ??? ?? ??? ?????? ??? ???? ????? ? ? ?? ????.

??????? ?? ??? ????? ????? ?? ?????? ? ? ??. ??? Magnus Hestenes? Eduard Stiefel? ?? ?????.

??? ??

??? ??? ?? ?? ?? ???? ?? ???? ????.

Ax = b

A? n x n ?????? (i.e. A^T = A), ?? ????? positive definite (i.e. Rⁿ?? ?? 0? ?? ??? x? ??? x^TAx > 0 )??, ????, x, b? n x 1 ?? ? ?????.

??? ??? ? ?? ??? ?? $\scriptstyle \mathbf {x} _{*}$?? ??.

??? ????? ?? ???

??? ??

$\mathbf {u} ^{\mathrm {T} }\mathbf {A} \mathbf {v} =0.$ ? ?,

??? ?? ?? ???? u ? v? ???? ??. (A? ???) A? ???? ?? ???? ?? ???? ??? ?? ?? ???? ?? ??.

$\langle \mathbf {u} ,\mathbf {v} \rangle _{\mathbf {A} }:=\langle \mathbf {A} \mathbf {u} ,\mathbf {v} \rangle =\langle \mathbf {u} ,\mathbf {A} ^{\mathrm {T} }\mathbf {v} \rangle =\langle \mathbf {u} ,\mathbf {A} \mathbf {v} \rangle =\mathbf {u} ^{\mathrm {T} }\mathbf {A} \mathbf {v} .$

?? ??? ??? ? ??? ??? ???? ???? ???? ??.

??? ???? ??????: ?? u? v? ???? v? u? ????.

$P=\{\mathbf {p} _{k}:\forall i\neq k,i,k\in [1,n],\langle \mathbf {p} _{i},\mathbf {p} _{k}\rangle _{A}=0\}$

? n? ???? ?? ???? ????? ????. ??? $P$? $\mathbb {R} ^{n}$? ????, ??? $P$??? ??? ? ? $\mathbf {Ax} =\mathbf {b}$? $\mathbf {x} _{*}$? ???? ? ??:

$\mathbf {x} _{*}=\sum _{i=1}^{n}\alpha _{i}\mathbf {p} _{i}$

??? ??? ??

$\mathbf {b} =\mathbf {A} \mathbf {x} _{*}=\sum _{i=1}^{n}\alpha _{i}\mathbf {A} \mathbf {p} _{i}.$

?? $\mathbf {p} _{k}\in P$? ???,

$\mathbf {p} _{k}^{\mathrm {T} }\mathbf {b} =\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {x} _{*}=\sum _{i=1}^{n}\alpha _{i}\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{i}=\alpha _{k}\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{k}.$

(???? $\forall i\neq k,p_{i},p_{k}$? ???? ???? ???)

$\alpha _{k}={\frac {\mathbf {p} _{k}^{\mathrm {T} }\mathbf {b} }{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{k}}}={\frac {\langle \mathbf {p} _{k},\mathbf {b} \rangle }{\,\,\,\langle \mathbf {p} _{k},\mathbf {p} _{k}\rangle _{\mathbf {A} }}}={\frac {\langle \mathbf {p} _{k},\mathbf {b} \rangle }{\,\,\,\|\mathbf {p} _{k}\|_{\mathbf {A} }^{2}}}.$

? ??? ?? ??? ??? ????? ? ??? ??? ?? ????.

??? Ax = b ???? ? ? ??? ??? ??. : n?? ?? ???? ??? ??, ??? ? ??? $\scriptstyle \alpha _{k}$? ????.

??? ????? ?? ???

?? ??? ?? ???? p_k ???? ????, ??? ? ? $\scriptstyle \mathbf {x} _{*}$ ? ?? ?? ?? ?? ??? ??? ??? ???? ?? ? ??. ??? ??? ???? ????? ?? ???? ??? ???. n? ?? ?? ???? ???? ??? ??? ?? ?? ??? ??? ?? ? ??? ?? ???? ?? ???.

$\scriptstyle \mathbf {x} _{*}$? ?? ???? x₀?? ??. ??? ???? ?? ?? x₀ = 0 (?????, ??? ? ?? Az = b ? Ax₀ ) ?? ??? ? ??. x₀? ?? ???? ??? ? ?? ??, ??? ???? ??? ? ? $\scriptstyle \mathbf {x} _{*}$(???? ????? ?? ?) ? ???? ???? ???? ??? ????. ? ??? ? ? $\scriptstyle \mathbf {x} _{*}$? ?? ??? ????? ???? ????? ????? ???. ??? ?? f(x)? ??? ??? ????? ??? ??? $\scriptstyle \mathbf {x} _{*}$? ??? ??? ?? ????.

$f(\mathbf {x} )={\frac {1}{2}}\mathbf {x} ^{\mathrm {T} }\mathbf {A} \mathbf {x} -\mathbf {x} ^{\mathrm {T} }\mathbf {b} ,\quad \mathbf {x} \in \mathbf {R} ^{n}.$

x = x₀?? f? ??(??:gradient)? ?? ?? ??? ?? ?? p₀? ???.

? f? ??(??:gradient)? Ax?b? ??, ?? ?x₀? ??? ????. (??? ?? ??? ?? ?? ?? ??? ??? ??? $\scriptstyle \mathbf {x} _{*}$ ? 0 ?? ? ??? x₀ 0?? ?? ??? ? ??. ) ??? p₀ = b?Ax₀?? ??. ? ???? ?? ???? ? ??(??:gradient)? ??? ? ???. ??? ? ??? ??? ?? ???(??:conjugate gradient method)??.

?? k?? ???? r_k? ??(??:residue)?? ??: $\mathbf {r} _{k}=\mathbf {b} -\mathbf {Ax} _{k}.\,$

r_k? x = x_k?? ?? f? ??(gradient)?? ?? ??? ??, ??? ?? ???? r_k ???? ??? ???. ??, ??? ? ??? p_k? ?? ???? ???? ????. ? ??? ??? ?????, ??? ?? ?? ?? ??? ??? ??? ??? ?? ?????? ??? ??? ??.

? ?? ????? ??-??? ?????? ???, ? ????? ?? -???? ????.

??? ??? ??? ??:

$\mathbf {p} _{k}=\mathbf {r} _{k}-\sum _{i<k}{\frac {\mathbf {p} _{i}^{\mathrm {T} }\mathbf {A} \mathbf {r} _{k}}{\mathbf {p} _{i}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{i}}}\mathbf {p} _{i}$

(??? ??? ????? ??? ?? ??? ??? ?? ??? ??) ? ??? ???, ? ?? ??? ??? ??? ?? ????.

$\mathbf {x} _{k+1}=\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k}$

$\alpha _{k}={\frac {\mathbf {p} _{k}^{\mathrm {T} }\mathbf {b} }{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{k}}}={\frac {\mathbf {p} _{k}^{\mathrm {T} }(\mathbf {r} _{k-1}+\mathbf {Ax} _{k-1})}{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{k}}}={\frac {\mathbf {p} _{k}^{\mathrm {T} }\mathbf {r} _{k-1}}{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {A} \mathbf {p} _{k}}},$

? ???

(p_k ?x_k-1 ? ???? ??? ??? ??? ????.)

? ??? ????

? ?? ????? ?? ???? ?? ??? ??? ????. ???, ? ????? ????? ??? ?? ???? ?? ???? ??? ??? ?? ??? ?? ?? ??? ???? ??? ??, ??? ????? ??? ?? ? ? ??. ??? ? ????? ??? ??? ?? ? ?? ? ? ??.

?? i < k? ??? r_k+1? p_i? ????. (?? ?? ???? ?? ??? ? ??) ???? ?? r_k, p_k, ??? x_k ? r_k+1, p_k+1, ??? x_k+1? ???? ??? ??. ??? ?? ??? ??-?? ??? ??? ???? ???? ????.

????, ?????? ?? ??? ??? A? ??? Ax = b? ?? ???? ??? ?? ??? ???? ??.

? ?? ?? x₀? ?? ??? ?? ?? 0? ? ? ??. ??? ?? ??? ??? ??? ?? ???.

${\begin{aligned}&\mathbf {r} _{0}:=\mathbf {b} -\mathbf {Ax} _{0}\\&\mathbf {p} _{0}:=\mathbf {r} _{0}\\&k:=0\\&{\hbox{repeat}}\\&\qquad \alpha _{k}:={\frac {\mathbf {r} _{k}^{\mathrm {T} }\mathbf {r} _{k}}{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {Ap} _{k}}}\\&\qquad \mathbf {x} _{k+1}:=\mathbf {x} _{k}+\alpha _{k}\mathbf {p} _{k}\\&\qquad \mathbf {r} _{k+1}:=\mathbf {r} _{k}-\alpha _{k}\mathbf {Ap} _{k}\\&\qquad {\hbox{if }}r_{k+1}{\hbox{ is sufficiently small then exit loop}}\\&\qquad \beta _{k}:={\frac {\mathbf {r} _{k+1}^{\mathrm {T} }\mathbf {r} _{k+1}}{\mathbf {r} _{k}^{\mathrm {T} }\mathbf {r} _{k}}}\\&\qquad \mathbf {p} _{k+1}:=\mathbf {r} _{k+1}+\beta _{k}\mathbf {p} _{k}\\&\qquad k:=k+1\\&{\hbox{end repeat}}\\&{\hbox{The result is }}\mathbf {x} _{k+1}\end{aligned}}$

??? ????? ?? ???? ??????. ?? ?? ??$\beta _{k}$? ??? ?? ???? ??? ????? ?? ????.

??? ??? ??

??????, $\alpha _{k}$ ? $\mathbf {r} _{k+1}$? $\mathbf {r} _{k}$? ??? ??? ????. ? ??? ??? ?? ?????.

$\alpha _{k}={\frac {\mathbf {r} _{k}^{\mathrm {T} }\mathbf {r} _{k}}{\mathbf {r} _{k}^{\mathrm {T} }\mathbf {Ap} _{k}}}={\frac {\mathbf {r} _{k}^{\mathrm {T} }\mathbf {r} _{k}}{\mathbf {p} _{k}^{\mathrm {T} }\mathbf {Ap} _{k}}}$

$\mathbf {r} _{k}=\mathbf {p} _{k}-\mathbf {\beta } _{k-1}\mathbf {p} _{k-1}$??? ? $\beta _{k}$? $\mathbf {p} _{k+1}$ ? $\mathbf {p} _{k}$? ??? ??? ????. ???, $\beta _{k}$ ?

$\beta _{k}=-{\frac {\mathbf {r} _{k+1}^{\mathrm {T} }A\mathbf {p} _{k}}{\mathbf {p} _{k}^{\mathrm {T} }A\mathbf {p} _{k}}}$

$\mathbf {r} _{k}=\mathbf {r} _{k-1}-\alpha _{k-1}A\mathbf {p} _{k-1}$? ????, ????? $A\mathbf {p} _{k-1}={\frac {1}{\alpha _{k-1}}}(\mathbf {r} _{k-1}-\mathbf {r} _{k})$, $\beta _{k}$? ??? ??? ?? ?? ??,

$\mathbf {r} _{k+1}^{\mathrm {T} }A\mathbf {p} _{k}={\frac {1}{\alpha _{k}}}\mathbf {r} _{k+1}^{\mathrm {T} }(\mathbf {r} _{k}-\mathbf {r} _{k+1})=-{\frac {1}{\alpha _{k}}}\mathbf {r} _{k+1}^{\mathrm {T} }\mathbf {r} _{k+1}$

$\mathbf {r} _{k+1}$ ? $\mathbf {r} _{k}$ ?? ??? ?? ???? ??? ???? ??? ?? ?? ??,

$\mathbf {p} _{k}^{\mathrm {T} }A\mathbf {p} _{k}=(\mathbf {r} _{k}+\beta _{k-1}\mathbf {p} _{k-1})^{\mathrm {T} }A\mathbf {p} _{k}={\frac {1}{\alpha _{k}}}\mathbf {r} _{k}^{\mathrm {T} }(\mathbf {r} _{k}-\mathbf {r} _{k+1})={\frac {1}{\alpha _{k}}}\mathbf {r} _{k}^{\mathrm {T} }\mathbf {r} _{k}$

?? ??? $\mathbf {p} _{k}$? ???? ?? ? ???? ???? ??? ???? ? ?????? $\alpha _{k}$? ??? ?$\beta$? ???.

?? (?????) ??

function [x] = conjgrad(A,b,x)
    r=b-A*x;
    p=r;
    rsold=r'*r;

    for i=1:1e6
        Ap=A*p;
        alpha=rsold/(p'*Ap);
        x=x+alpha*p;
        r=r-alpha*Ap;
        rsnew=r'*r;
        if sqrt(rsnew)<1e-10
              break;
        end
        p=r+rsnew/rsold*p;
        rsold=rsnew;
    end
end

??? ??

?? ???? ???? ???, ??? ??? ?? ???. ??? ?? ?? ?? Ax = b? ????

$\mathbf {A} \mathbf {x} ={\begin{bmatrix}4&1\\1&3\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}={\begin{bmatrix}1\\2\end{bmatrix}},$

? ?? ?? ??? ?? ?? ??? ??? ?? ???? ???? ?? ???? ? ???? ??? ???.

$\mathbf {x} _{0}={\begin{bmatrix}2\\1\end{bmatrix}}$

?

??? ??? ??? ?? :: $\mathbf {x} ={\begin{bmatrix}{\frac {1}{11}}\\\\{\frac {7}{11}}\end{bmatrix}}$

??? ??? ??? x₀? ??? ?? ?? r₀? ?? ?? ???.

? ??? ? r₀ = b - Ax₀???? ????, ??? ??? ???? ??? ??.

$\mathbf {r} _{0}={\begin{bmatrix}1\\2\end{bmatrix}}-{\begin{bmatrix}4&1\\1&3\end{bmatrix}}{\begin{bmatrix}2\\1\end{bmatrix}}={\begin{bmatrix}-8\\-3\end{bmatrix}}.$

??? ??? ?? ???? ???, ??? ??? ?? ????p₀ ??? ?? ?? r₀? ??? ???.

???? ?? ???? p_k? ???? ??? ?? ???.

??? ?? ?? ??? ???? ?? ??? α₀? ????.

$\alpha _{0}={\frac {\mathbf {r} _{0}^{\mathrm {T} }\mathbf {r} _{0}}{\mathbf {p} _{0}^{\mathrm {T} }\mathbf {Ap} _{0}}}={\frac {{\begin{bmatrix}-8&-3\end{bmatrix}}{\begin{bmatrix}-8\\-3\end{bmatrix}}}{{\begin{bmatrix}-8&-3\end{bmatrix}}{\begin{bmatrix}4&1\\1&3\end{bmatrix}}{\begin{bmatrix}-8\\-3\end{bmatrix}}}}={\frac {73}{331}}.$

??? ? ?? ???? x₁? ?? ? ? ??.

$\mathbf {x} _{1}=\mathbf {x} _{0}+\alpha _{0}\mathbf {p} _{0}={\begin{bmatrix}2\\1\end{bmatrix}}+{\frac {73}{331}}{\begin{bmatrix}-8\\-3\end{bmatrix}}={\begin{bmatrix}0.2356\\0.3384\end{bmatrix}}.$

? ??? ??? ?? ??? ????, ? ?? "???" ?? ??? ?x₁? ??. ??? ?? ???? ? ?? ???? ??? ?? ?? r₁? ??? ???.

$\mathbf {r} _{1}=\mathbf {r} _{0}-\alpha _{0}\mathbf {A} \mathbf {p} _{0}={\begin{bmatrix}-8\\-3\end{bmatrix}}-{\frac {73}{331}}{\begin{bmatrix}4&1\\1&3\end{bmatrix}}{\begin{bmatrix}-8\\-3\end{bmatrix}}={\begin{bmatrix}-0.2810\\0.7492\end{bmatrix}}.$

? ????? ??? ?? ??? ?? ?? ?? p₁? ????? ????? ??? ??? β₀? ???? ???.

$\beta _{0}={\frac {\mathbf {r} _{1}^{\mathrm {T} }\mathbf {r} _{1}}{\mathbf {r} _{0}^{\mathrm {T} }\mathbf {r} _{0}}}={\frac {{\begin{bmatrix}-0.2810&0.7492\end{bmatrix}}{\begin{bmatrix}-0.2810\\0.7492\end{bmatrix}}}{{\begin{bmatrix}-8&-3\end{bmatrix}}{\begin{bmatrix}-8\\-3\end{bmatrix}}}}=0.0088.$

??, ? ??? β₀? ????, ??? ? ??? ???? ?? ?? ?? p₁? ?? ? ? ??.

$\mathbf {p} _{1}=\mathbf {r} _{1}+\beta _{0}\mathbf {p} _{0}={\begin{bmatrix}-0.2810\\0.7492\end{bmatrix}}+0.0088{\begin{bmatrix}-8\\-3\end{bmatrix}}={\begin{bmatrix}-0.3511\\0.7229\end{bmatrix}}.$

??? α₀?? ???? ?? ???? ?? ?? p₁? ???? ?? ??? α₁? ?? ? ? ??.

$\alpha _{1}={\frac {\mathbf {r} _{1}^{\mathrm {T} }\mathbf {r} _{1}}{\mathbf {p} _{1}^{\mathrm {T} }\mathbf {Ap} _{1}}}={\frac {{\begin{bmatrix}-0.2810&0.7492\end{bmatrix}}{\begin{bmatrix}-0.2810\\0.7492\end{bmatrix}}}{{\begin{bmatrix}-0.3511&0.7229\end{bmatrix}}{\begin{bmatrix}4&1\\1&3\end{bmatrix}}{\begin{bmatrix}-0.3511\\0.7229\end{bmatrix}}}}=0.4122.$

?????, ??? x₁? ??? ?? ???? x₂? ???.

$\mathbf {x} _{2}=\mathbf {x} _{1}+\alpha _{1}\mathbf {p} _{1}={\begin{bmatrix}0.2356\\0.3384\end{bmatrix}}+0.4122{\begin{bmatrix}-0.3511\\0.7229\end{bmatrix}}={\begin{bmatrix}0.0909\\0.6364\end{bmatrix}}.$

? ??, x₂? ? ?? ?? x₁ ? x₀?? "???" ???? ??.

?? ??? ??? ??? ? ??? ??? ??? ????, ? ??? ?? ????? n=2 ???? ? ?? ? ???. (n? ? ?? ???).

?? ??

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