{"id":110129,"date":"2026-09-26T06:49:04","date_gmt":"2026-09-25T22:49:04","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/110129.html"},"modified":"2026-09-26T06:49:04","modified_gmt":"2026-09-25T22:49:04","slug":"c%e4%bb%a3%e7%a0%81%e5%ae%9e%e7%8e%b0matlab%e4%b8%ad%e7%9a%84lsqnonlin%e5%87%bd%e6%95%b0%e5%8a%9f%e8%83%bd","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/110129.html","title":{"rendered":"C++\u4ee3\u7801\u5b9e\u73b0MATLAB\u4e2d\u7684lsqnonlin\u51fd\u6570\u529f\u80fd"},"content":{"rendered":"

\/\/ =============================================================<\/span>
\n\/\/ lsqnonlin.cpp<\/span>
\n\/\/ MATLAB \u9ed8\u8ba4\u7b97\u6cd5 trust-region-reflective \u7684\u72ec\u7acb\u5b9e\u73b0<\/span>
\n\/\/ \u7f16\u8bd1: g++ -O2 -std=c++17 lsqnonlin_trr.cpp -o lsqtrr<\/span>
\n\/\/ =============================================================<\/span>
\n#<\/span>include<\/span> <cstdio><\/span><\/span>
\n#<\/span>include<\/span> <cmath><\/span><\/span>
\n#<\/span>include<\/span> <vector><\/span><\/span>
\n#<\/span>include<\/span> <functional><\/span><\/span>
\n#<\/span>include<\/span> <algorithm><\/span><\/span>
\n#<\/span>include<\/span> <limits><\/span><\/span><\/p>\n

using<\/span> Vec =<\/span> std::<\/span>vector<<\/span>double<\/span>><\/span>;<\/span><\/p>\n

struct<\/span> Matrix<\/span> {<\/span>
\n int<\/span> rows =<\/span> 0<\/span>,<\/span> cols =<\/span> 0<\/span>;<\/span>
\n std::<\/span>vector<<\/span>double<\/span>><\/span> a;<\/span>
\n Matrix<\/span>(<\/span>)<\/span> =<\/span> default<\/span>;<\/span>
\n Matrix<\/span>(<\/span>int<\/span> r,<\/span> int<\/span> c)<\/span> :<\/span> rows<\/span>(<\/span>r)<\/span>,<\/span> cols<\/span>(<\/span>c)<\/span>,<\/span> a<\/span>(<\/span>(<\/span>size_t)<\/span>r*<\/span>c,<\/span> 0.0<\/span>)<\/span> {<\/span>}<\/span>
\n double<\/span>&<\/span> operator<\/span>(<\/span>)<\/span>(<\/span>int<\/span> i,<\/span> int<\/span> j)<\/span> {<\/span> return<\/span> a[<\/span>(<\/span>size_t)<\/span>i*<\/span>cols +<\/span> j]<\/span>;<\/span> }<\/span>
\n double<\/span> operator<\/span>(<\/span>)<\/span>(<\/span>int<\/span> i,<\/span> int<\/span> j)<\/span> const<\/span> {<\/span> return<\/span> a[<\/span>(<\/span>size_t)<\/span>i*<\/span>cols +<\/span> j]<\/span>;<\/span> }<\/span>
\n}<\/span>;<\/span><\/p>\n

static<\/span> double<\/span> dot<\/span>(<\/span>const<\/span> Vec&<\/span> a,<\/span> const<\/span> Vec&<\/span> b)<\/span> {<\/span>
\n double<\/span> s =<\/span> 0.0<\/span>;<\/span> for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> a.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span> s +=<\/span> a[<\/span>i]<\/span>*<\/span>b[<\/span>i]<\/span>;<\/span> return<\/span> s;<\/span>
\n}<\/span>
\nstatic<\/span> double<\/span> nrm2<\/span>(<\/span>const<\/span> Vec&<\/span> a)<\/span> {<\/span> return<\/span> dot<\/span>(<\/span>a,<\/span> a)<\/span>;<\/span> }<\/span>
\nstatic<\/span> double<\/span> norm<\/span>(<\/span>const<\/span> Vec&<\/span> a)<\/span> {<\/span> return<\/span> std::<\/span>sqrt<\/span>(<\/span>dot<\/span>(<\/span>a,<\/span> a)<\/span>)<\/span>;<\/span> }<\/span><\/p>\n

\/\/ ———- LDLT \u6c42\u89e3\u5bf9\u79f0\u7cfb\u7edf A x = b ———-<\/span>
\nstatic<\/span> bool<\/span> ldltSolve<\/span>(<\/span>const<\/span> Matrix&<\/span> A,<\/span> const<\/span> Vec&<\/span> b,<\/span> Vec&<\/span> x)<\/span> {<\/span>
\n const<\/span> int<\/span> n =<\/span> A.<\/span>rows;<\/span>
\n Matrix L<\/span>(<\/span>n,<\/span> n)<\/span>;<\/span>
\n Vec D<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> 0<\/span>;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> {<\/span>
\n double<\/span> d =<\/span> A<\/span>(<\/span>j,<\/span> j)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> j;<\/span> ++<\/span>k)<\/span> d -=<\/span> L<\/span>(<\/span>j,<\/span> k)<\/span>*<\/span>L<\/span>(<\/span>j,<\/span> k)<\/span>*<\/span>D[<\/span>k]<\/span>;<\/span>
\n if<\/span> (<\/span>!<\/span>std::<\/span>isfinite<\/span>(<\/span>d)<\/span> ||<\/span> std::<\/span>abs<\/span>(<\/span>d)<\/span> <<\/span> 1e-300<\/span>)<\/span> return<\/span> false<\/span>;<\/span>
\n D[<\/span>j]<\/span> =<\/span> d;<\/span> L<\/span>(<\/span>j,<\/span> j)<\/span> =<\/span> 1.0<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> j+<\/span>1<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n double<\/span> s =<\/span> A<\/span>(<\/span>i,<\/span> j)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> j;<\/span> ++<\/span>k)<\/span> s -=<\/span> L<\/span>(<\/span>i,<\/span> k)<\/span>*<\/span>L<\/span>(<\/span>j,<\/span> k)<\/span>*<\/span>D[<\/span>k]<\/span>;<\/span>
\n L<\/span>(<\/span>i,<\/span> j)<\/span> =<\/span> s \/<\/span> D[<\/span>j]<\/span>;<\/span>
\n if<\/span> (<\/span>!<\/span>std::<\/span>isfinite<\/span>(<\/span>L<\/span>(<\/span>i,<\/span> j)<\/span>)<\/span>)<\/span> return<\/span> false<\/span>;<\/span>
\n }<\/span>
\n }<\/span>
\n Vec y<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n double<\/span> s =<\/span> b[<\/span>i]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> i;<\/span> ++<\/span>k)<\/span> s -=<\/span> L<\/span>(<\/span>i,<\/span> k)<\/span>*<\/span>y[<\/span>k]<\/span>;<\/span>
\n y[<\/span>i]<\/span> =<\/span> s;<\/span>
\n }<\/span>
\n Vec z<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> z[<\/span>i]<\/span> =<\/span> y[<\/span>i]<\/span>\/<\/span>D[<\/span>i]<\/span>;<\/span>
\n x.<\/span>assign<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> n–<\/span>1<\/span>;<\/span> i >=<\/span> 0<\/span>;<\/span> —<\/span>i)<\/span> {<\/span>
\n double<\/span> s =<\/span> z[<\/span>i]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> i+<\/span>1<\/span>;<\/span> k <<\/span> n;<\/span> ++<\/span>k)<\/span> s -=<\/span> L<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>x[<\/span>k]<\/span>;<\/span>
\n x[<\/span>i]<\/span> =<\/span> s;<\/span>
\n }<\/span>
\n for<\/span> (<\/span>double<\/span> v :<\/span> x)<\/span> if<\/span> (<\/span>!<\/span>std::<\/span>isfinite<\/span>(<\/span>v)<\/span>)<\/span> return<\/span> false<\/span>;<\/span>
\n return<\/span> true<\/span>;<\/span>
\n}<\/span><\/p>\n

\/\/ ———- \u8ba1\u7b97 JtJ \u4e0e Jtf ———-<\/span>
\nstatic<\/span> void<\/span> jtj_jtf<\/span>(<\/span>const<\/span> Matrix&<\/span> J,<\/span> const<\/span> Vec&<\/span> f,<\/span> Matrix&<\/span> JtJ,<\/span> Vec&<\/span> Jtf)<\/span> {<\/span>
\n int<\/span> m =<\/span> J.<\/span>rows,<\/span> n =<\/span> J.<\/span>cols;<\/span>
\n JtJ =<\/span> Matrix<\/span>(<\/span>n,<\/span> n)<\/span>;<\/span>
\n Jtf.<\/span>assign<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> {<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> m;<\/span> ++<\/span>k)<\/span> Jtf[<\/span>i]<\/span> +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>f[<\/span>k]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> i;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> {<\/span>
\n double<\/span> s =<\/span> 0.0<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> m;<\/span> ++<\/span>k)<\/span> s +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>J<\/span>(<\/span>k,<\/span> j)<\/span>;<\/span>
\n JtJ<\/span>(<\/span>i,<\/span> j)<\/span> =<\/span> JtJ<\/span>(<\/span>j,<\/span> i)<\/span> =<\/span> s;<\/span>
\n }<\/span>
\n }<\/span>
\n}<\/span><\/p>\n

\/\/ ———- \u9ad8\u65af-\u725b\u987f\u6b65 ———-<\/span>
\nstatic<\/span> bool<\/span> gaussNewtonStep<\/span>(<\/span>const<\/span> Matrix&<\/span> J,<\/span> const<\/span> Vec&<\/span> f,<\/span> Vec&<\/span> s)<\/span> {<\/span>
\n Matrix JtJ;<\/span> Vec Jtf;<\/span>
\n jtj_jtf<\/span>(<\/span>J,<\/span> f,<\/span> JtJ,<\/span> Jtf)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> JtJ.<\/span>rows;<\/span> ++<\/span>i)<\/span> JtJ<\/span>(<\/span>i,<\/span> i)<\/span> +=<\/span> 1e-14<\/span>;<\/span>
\n Vec rhs<\/span>(<\/span>JtJ.<\/span>rows)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> JtJ.<\/span>rows;<\/span> ++<\/span>i)<\/span> rhs[<\/span>i]<\/span> =<\/span> –<\/span>Jtf[<\/span>i]<\/span>;<\/span>
\n return<\/span> ldltSolve<\/span>(<\/span>JtJ,<\/span> rhs,<\/span> s)<\/span>;<\/span>
\n}<\/span><\/p>\n

\/\/ ———- Dogleg \u5b50\u95ee\u9898：\u5728 \u2016s\u2016 \u2264 \u0394 \u5185 min \u2016J s + f\u2016 ———-<\/span>
\nstatic<\/span> Vec solveTrustRegion<\/span>(<\/span>const<\/span> Matrix&<\/span> J,<\/span> const<\/span> Vec&<\/span> f,<\/span> double<\/span> Delta)<\/span> {<\/span>
\n const<\/span> int<\/span> m =<\/span> J.<\/span>rows,<\/span> n =<\/span> J.<\/span>cols;<\/span>
\n Vec s_gn;<\/span>
\n bool<\/span> okGN =<\/span> gaussNewtonStep<\/span>(<\/span>J,<\/span> f,<\/span> s_gn)<\/span>;<\/span>
\n if<\/span> (<\/span>okGN &&<\/span> norm<\/span>(<\/span>s_gn)<\/span> <=<\/span> Delta)<\/span> return<\/span> s_gn;<\/span> \/\/ GN \u6b65\u5728\u57df\u5185<\/span><\/p>\n

\/\/ Cauchy \u70b9<\/span>
\n Vec g<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> m;<\/span> ++<\/span>k)<\/span> g[<\/span>i]<\/span> +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>f[<\/span>k]<\/span>;<\/span>
\n double<\/span> gg =<\/span> dot<\/span>(<\/span>g,<\/span> g)<\/span>;<\/span>
\n if<\/span> (<\/span>gg <<\/span> 1e-300<\/span>)<\/span> return<\/span> Vec<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span><\/p>\n

Vec Jg<\/span>(<\/span>m,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> m;<\/span> ++<\/span>k)<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> Jg[<\/span>k]<\/span> +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>g[<\/span>i]<\/span>;<\/span>
\n double<\/span> JgJg =<\/span> dot<\/span>(<\/span>Jg,<\/span> Jg)<\/span>;<\/span>
\n if<\/span> (<\/span>JgJg <<\/span> 1e-300<\/span>)<\/span> return<\/span> Vec<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span><\/p>\n

double<\/span> alpha =<\/span> gg \/<\/span> JgJg;<\/span>
\n Vec s_sd<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> s_sd[<\/span>i]<\/span> =<\/span> –<\/span>alpha *<\/span> g[<\/span>i]<\/span>;<\/span><\/p>\n

double<\/span> nsd =<\/span> norm<\/span>(<\/span>s_sd)<\/span>;<\/span>
\n if<\/span> (<\/span>nsd >=<\/span> Delta ||<\/span> !<\/span>okGN)<\/span> {<\/span> \/\/ SD \u6b65\u51fa\u754c（\u6216 GN \u4e0d\u53ef\u7528）\u2192 \u7f29\u653e\u5230\u8fb9\u754c<\/span>
\n double<\/span> sc =<\/span> Delta \/<\/span> nsd;<\/span>
\n Vec s<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> s[<\/span>i]<\/span> =<\/span> sc *<\/span> s_sd[<\/span>i]<\/span>;<\/span>
\n return<\/span> s;<\/span>
\n }<\/span><\/p>\n

\/\/ Dogleg \u63d2\u503c：\u03c4 \u4f7f \u2016s_sd + \u03c4(s_gn – s_sd)\u2016 = \u0394<\/span>
\n Vec d<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> d[<\/span>i]<\/span> =<\/span> s_gn[<\/span>i]<\/span> –<\/span> s_sd[<\/span>i]<\/span>;<\/span>
\n double<\/span> a2 =<\/span> dot<\/span>(<\/span>d,<\/span> d)<\/span>;<\/span>
\n double<\/span> b2 =<\/span> 2.0<\/span> *<\/span> dot<\/span>(<\/span>s_sd,<\/span> d)<\/span>;<\/span>
\n double<\/span> c2 =<\/span> dot<\/span>(<\/span>s_sd,<\/span> s_sd)<\/span> –<\/span> Delta*<\/span>Delta;<\/span>
\n double<\/span> disc =<\/span> std::<\/span>max<\/span>(<\/span>0.0<\/span>,<\/span> b2*<\/span>b2 –<\/span> 4<\/span>*<\/span>a2*<\/span>c2)<\/span>;<\/span>
\n double<\/span> tau =<\/span> (<\/span>a2 ><\/span> 0<\/span>)<\/span> ?<\/span> (<\/span>–<\/span>b2 +<\/span> std::<\/span>sqrt<\/span>(<\/span>disc)<\/span>)<\/span> \/<\/span> (<\/span>2<\/span>*<\/span>a2)<\/span> :<\/span> 1.0<\/span>;<\/span>
\n tau =<\/span> std::<\/span>max<\/span>(<\/span>0.0<\/span>,<\/span> std::<\/span>min<\/span>(<\/span>1.0<\/span>,<\/span> tau)<\/span>)<\/span>;<\/span>
\n Vec s<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> s[<\/span>i]<\/span> =<\/span> s_sd[<\/span>i]<\/span> +<\/span> tau *<\/span> d[<\/span>i]<\/span>;<\/span>
\n return<\/span> s;<\/span>
\n}<\/span><\/p>\n

\/\/ ==================== lsqnonlin (TRR) ====================<\/span>
\nclass<\/span> Lsqnonlin<\/span> {<\/span>
\npublic<\/span>:<\/span>
\n using<\/span> FunType =<\/span> std::<\/span>function<<\/span>Vec<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span>><\/span>;<\/span>
\n using<\/span> JacType =<\/span> std::<\/span>function<<\/span>Matrix<\/span>(<\/span>const<\/span> Vec&<\/span>)<\/span>><\/span>;<\/span><\/p>\n

struct<\/span> Options<\/span> {<\/span>
\n int<\/span> maxIter =<\/span> 400<\/span>;<\/span>
\n double<\/span> stepTol =<\/span> 1e-8<\/span>;<\/span>
\n double<\/span> funcTol =<\/span> 1e-12<\/span>;<\/span>
\n double<\/span> optimalityTol =<\/span> 1e-8<\/span>;<\/span>
\n double<\/span> Delta0 =<\/span> 1.0<\/span>;<\/span> \/\/ \u521d\u59cb\u4fe1\u8d56\u57df\u534a\u5f84<\/span>
\n bool<\/span> verbose =<\/span> false<\/span>;<\/span>
\n }<\/span>;<\/span><\/p>\n

struct<\/span> Result<\/span> {<\/span>
\n Vec x,<\/span> residual;<\/span>
\n double<\/span> cost =<\/span> 0.0<\/span>;<\/span>
\n int<\/span> iterations =<\/span> 0<\/span>;<\/span>
\n int<\/span> exitFlag =<\/span> 0<\/span>;<\/span> \/\/ 1:\u6700\u4f18\u6027\u6536\u655b 2:\u6b65\u957f\u6536\u655b 3:\u4fe1\u8d56\u57df\u8fc7\u5c0f 0:\u6700\u5927\u8fed\u4ee3<\/span>
\n bool<\/span> success<\/span>(<\/span>)<\/span> const<\/span> {<\/span> return<\/span> exitFlag ><\/span> 0<\/span>;<\/span> }<\/span>
\n }<\/span>;<\/span><\/p>\n

explicit<\/span> Lsqnonlin<\/span>(<\/span>FunType fun)<\/span> :<\/span> fun_<\/span>(<\/span>std::<\/span>move<\/span>(<\/span>fun)<\/span>)<\/span> {<\/span>}<\/span>
\n Lsqnonlin<\/span>(<\/span>FunType fun,<\/span> JacType jac)<\/span> :<\/span> fun_<\/span>(<\/span>std::<\/span>move<\/span>(<\/span>fun)<\/span>)<\/span>,<\/span> jac_<\/span>(<\/span>std::<\/span>move<\/span>(<\/span>jac)<\/span>)<\/span> {<\/span>}<\/span><\/p>\n

Result solve<\/span>(<\/span>const<\/span> Vec&<\/span> x0)<\/span> {<\/span> return<\/span> solve<\/span>(<\/span>x0,<\/span> Options<\/span>(<\/span>)<\/span>)<\/span>;<\/span> }<\/span>
\n Result solve<\/span>(<\/span>const<\/span> Vec&<\/span> x0,<\/span> const<\/span> Options&<\/span> opts)<\/span> {<\/span>
\n const<\/span> int<\/span> n =<\/span> (<\/span>int<\/span>)<\/span>x0.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
\n Result res;<\/span>
\n Vec x =<\/span> x0;<\/span><\/p>\n

Vec f =<\/span> fun_<\/span>(<\/span>x)<\/span>;<\/span>
\n double<\/span> cost =<\/span> 0.5<\/span> *<\/span> nrm2<\/span>(<\/span>f)<\/span>;<\/span>
\n Matrix J =<\/span> computeJacobian<\/span>(<\/span>x,<\/span> f)<\/span>;<\/span><\/p>\n

double<\/span> Delta =<\/span> opts.<\/span>Delta0;<\/span>
\n const<\/span> double<\/span> eta =<\/span> 0.15<\/span>;<\/span> \/\/ \u6b65\u957f\u63a5\u53d7\u9608\u503c<\/span><\/p>\n

int<\/span> iter =<\/span> 0<\/span>;<\/span>
\n for<\/span> (<\/span>iter =<\/span> 0<\/span>;<\/span> iter <<\/span> opts.<\/span>maxIter;<\/span> ++<\/span>iter)<\/span> {<\/span>
\n \/\/ \u4e00\u9636\u6700\u4f18\u6027（\u65e0\u7a77\u8303\u6570 \u2016J\u1d40f\u2016\u221e）<\/span>
\n Vec g<\/span>(<\/span>n,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> (<\/span>int<\/span>)<\/span>f.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>k)<\/span>
\n g[<\/span>i]<\/span> +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>f[<\/span>k]<\/span>;<\/span>
\n double<\/span> gInf =<\/span> 0.0<\/span>;<\/span>
\n for<\/span> (<\/span>double<\/span> v :<\/span> g)<\/span> gInf =<\/span> std::<\/span>max<\/span>(<\/span>gInf,<\/span> std::<\/span>abs<\/span>(<\/span>v)<\/span>)<\/span>;<\/span><\/p>\n

if<\/span> (<\/span>gInf <<\/span> opts.<\/span>optimalityTol)<\/span> {<\/span> res.<\/span>exitFlag =<\/span> 1<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n if<\/span> (<\/span>Delta <<\/span> opts.<\/span>stepTol)<\/span> {<\/span> res.<\/span>exitFlag =<\/span> 3<\/span>;<\/span> break<\/span>;<\/span> }<\/span><\/p>\n

\/\/ \u5b50\u95ee\u9898<\/span>
\n Vec s =<\/span> solveTrustRegion<\/span>(<\/span>J,<\/span> f,<\/span> Delta)<\/span>;<\/span><\/p>\n

\/\/ \u9884\u6d4b\u4e0b\u964d<\/span>
\n Vec Jsf<\/span>(<\/span>f.<\/span>size<\/span>(<\/span>)<\/span>,<\/span> 0.0<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> k =<\/span> 0<\/span>;<\/span> k <<\/span> (<\/span>int<\/span>)<\/span>f.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>k)<\/span> {<\/span>
\n double<\/span> v =<\/span> f[<\/span>k]<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> v +=<\/span> J<\/span>(<\/span>k,<\/span> i)<\/span>*<\/span>s[<\/span>i]<\/span>;<\/span>
\n Jsf[<\/span>k]<\/span> =<\/span> v;<\/span>
\n }<\/span>
\n double<\/span> predRed =<\/span> 0.5<\/span> *<\/span> (<\/span>nrm2<\/span>(<\/span>f)<\/span> –<\/span> nrm2<\/span>(<\/span>Jsf)<\/span>)<\/span>;<\/span><\/p>\n

\/\/ \u8bd5\u9a8c\u70b9<\/span>
\n Vec xNew<\/span>(<\/span>n)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> n;<\/span> ++<\/span>i)<\/span> xNew[<\/span>i]<\/span> =<\/span> x[<\/span>i]<\/span> +<\/span> s[<\/span>i]<\/span>;<\/span>
\n Vec fNew =<\/span> fun_<\/span>(<\/span>xNew)<\/span>;<\/span>
\n double<\/span> costNew =<\/span> 0.5<\/span> *<\/span> nrm2<\/span>(<\/span>fNew)<\/span>;<\/span><\/p>\n

double<\/span> actRed =<\/span> cost –<\/span> costNew;<\/span>
\n double<\/span> rho =<\/span> (<\/span>predRed ><\/span> 0<\/span>)<\/span> ?<\/span> actRed \/<\/span> predRed :<\/span> –<\/span>1.0<\/span>;<\/span><\/p>\n

if<\/span> (<\/span>opts.<\/span>verbose)<\/span> {<\/span>
\n std::<\/span>printf<\/span>(<\/span>" Iter %3d: cost=%.10e rho=%+.4e Delta=%.4e |g|=%.4e\\\\n"<\/span>,<\/span>
\n iter,<\/span> cost,<\/span> rho,<\/span> Delta,<\/span> gInf)<\/span>;<\/span>
\n }<\/span><\/p>\n

if<\/span> (<\/span>rho ><\/span> eta)<\/span> {<\/span>
\n double<\/span> stepNorm =<\/span> norm<\/span>(<\/span>s)<\/span>;<\/span>
\n double<\/span> costRel =<\/span> std::<\/span>abs<\/span>(<\/span>cost –<\/span> costNew)<\/span> \/<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> cost)<\/span>;<\/span><\/p>\n

x =<\/span> xNew;<\/span> f =<\/span> fNew;<\/span> cost =<\/span> costNew;<\/span>
\n J =<\/span> computeJacobian<\/span>(<\/span>x,<\/span> f)<\/span>;<\/span><\/p>\n

if<\/span> (<\/span>rho ><\/span> 0.75<\/span>)<\/span> Delta =<\/span> std::<\/span>min<\/span>(<\/span>2.0<\/span>*<\/span>Delta,<\/span> 1e10<\/span>)<\/span>;<\/span>
\n else<\/span> if<\/span> (<\/span>rho <<\/span> 0.25<\/span>)<\/span> Delta *=<\/span> 0.5<\/span>;<\/span><\/p>\n

if<\/span> (<\/span>stepNorm <<\/span> opts.<\/span>stepTol *<\/span> (<\/span>norm<\/span>(<\/span>x)<\/span> +<\/span> opts.<\/span>stepTol)<\/span>)<\/span> {<\/span>
\n res.<\/span>exitFlag =<\/span> 2<\/span>;<\/span> break<\/span>;<\/span>
\n }<\/span>
\n if<\/span> (<\/span>costRel <<\/span> opts.<\/span>funcTol)<\/span> {<\/span> res.<\/span>exitFlag =<\/span> 1<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n }<\/span> else<\/span> {<\/span>
\n Delta *=<\/span> 0.5<\/span>;<\/span>
\n if<\/span> (<\/span>Delta <<\/span> opts.<\/span>stepTol)<\/span> {<\/span> res.<\/span>exitFlag =<\/span> 3<\/span>;<\/span> break<\/span>;<\/span> }<\/span>
\n }<\/span>
\n }<\/span><\/p>\n

res.<\/span>x =<\/span> x;<\/span> res.<\/span>residual =<\/span> f;<\/span> res.<\/span>cost =<\/span> cost;<\/span> res.<\/span>iterations =<\/span> iter;<\/span>
\n return<\/span> res;<\/span>
\n }<\/span><\/p>\n

private<\/span>:<\/span>
\n Matrix computeJacobian<\/span>(<\/span>const<\/span> Vec&<\/span> x,<\/span> const<\/span> Vec&<\/span> f)<\/span> const<\/span> {<\/span>
\n return<\/span> jac_ ?<\/span> jac_<\/span>(<\/span>x)<\/span> :<\/span> numericalJacobian<\/span>(<\/span>x,<\/span> f)<\/span>;<\/span>
\n }<\/span>
\n Matrix numericalJacobian<\/span>(<\/span>const<\/span> Vec&<\/span> x,<\/span> const<\/span> Vec&<\/span> f0)<\/span> const<\/span> {<\/span>
\n int<\/span> n =<\/span> (<\/span>int<\/span>)<\/span>x.<\/span>size<\/span>(<\/span>)<\/span>,<\/span> m =<\/span> (<\/span>int<\/span>)<\/span>f0.<\/span>size<\/span>(<\/span>)<\/span>;<\/span>
\n Matrix J<\/span>(<\/span>m,<\/span> n)<\/span>;<\/span>
\n double<\/span> sqEps =<\/span> std::<\/span>sqrt<\/span>(<\/span>std::<\/span>numeric_limits<\/span><<\/span>double<\/span>><\/span>::<\/span>epsilon<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> j =<\/span> 0<\/span>;<\/span> j <<\/span> n;<\/span> ++<\/span>j)<\/span> {<\/span>
\n double<\/span> h =<\/span> sqEps *<\/span> std::<\/span>max<\/span>(<\/span>1.0<\/span>,<\/span> std::<\/span>abs<\/span>(<\/span>x[<\/span>j]<\/span>)<\/span>)<\/span>;<\/span>
\n Vec xp =<\/span> x;<\/span> xp[<\/span>j]<\/span> +=<\/span> h;<\/span>
\n Vec fp =<\/span> fun_<\/span>(<\/span>xp)<\/span>;<\/span>
\n for<\/span> (<\/span>int<\/span> i =<\/span> 0<\/span>;<\/span> i <<\/span> m;<\/span> ++<\/span>i)<\/span> J<\/span>(<\/span>i,<\/span> j)<\/span> =<\/span> (<\/span>fp[<\/span>i]<\/span> –<\/span> f0[<\/span>i]<\/span>)<\/span> \/<\/span> h;<\/span>
\n }<\/span>
\n return<\/span> J;<\/span>
\n }<\/span>
\n FunType fun_;<\/span>
\n JacType jac_;<\/span>
\n}<\/span>;<\/span><\/p>\n

\/\/ ==================== \u6d4b\u8bd5\u4e3b\u7a0b\u5e8f ====================<\/span>
\nstatic<\/span> void<\/span> printVec<\/span>(<\/span>const<\/span> char<\/span>*<\/span> name,<\/span> const<\/span> Vec&<\/span> v)<\/span> {<\/span>
\n std::<\/span>printf<\/span>(<\/span>"%s = ["<\/span>,<\/span> name)<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> v.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span>
\n std::<\/span>printf<\/span>(<\/span>"%s%.12g"<\/span>,<\/span> i ?<\/span> ", "<\/span> :<\/span> ""<\/span>,<\/span> v[<\/span>i]<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>"]\\\\n"<\/span>)<\/span>;<\/span>
\n}<\/span><\/p>\n

int<\/span> main<\/span>(<\/span>)<\/span> {<\/span>
\n std::<\/span>printf<\/span>(<\/span>"=========================================================\\\\n"<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" MATLAB lsqnonlin \u9ed8\u8ba4\u7b97\u6cd5: trust-region-reflective\\\\n"<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>"=========================================================\\\\n\\\\n"<\/span>)<\/span>;<\/span><\/p>\n

\/\/ ———- \u7528\u4f8b 1: Rosenbrock \u6700\u5c0f\u4e8c\u4e58 ———-<\/span>
\n {<\/span>
\n std::<\/span>printf<\/span>(<\/span>"———- \u7528\u4f8b 1: Rosenbrock \u6700\u5c0f\u4e8c\u4e58 ———-\\\\n"<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" r1 = 10*(x2 – x1^2)\\\\n"<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" r2 = 1 – x1\\\\n"<\/span>)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" x0 = [-1.2, 1.0]，\u671f\u671b\u6700\u4f18 x* = [1, 1]\\\\n\\\\n"<\/span>)<\/span>;<\/span><\/p>\n

auto<\/span> fun =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Vec {<\/span>
\n return<\/span> {<\/span> 10.0<\/span>*<\/span>(<\/span>x[<\/span>1<\/span>]<\/span> –<\/span> x[<\/span>0<\/span>]<\/span>*<\/span>x[<\/span>0<\/span>]<\/span>)<\/span>,<\/span> 1.0<\/span> –<\/span> x[<\/span>0<\/span>]<\/span> }<\/span>;<\/span>
\n }<\/span>;<\/span>
\n auto<\/span> jac =<\/span> [<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> x)<\/span> -><\/span> Matrix {<\/span>
\n Matrix J<\/span>(<\/span>2<\/span>,<\/span> 2<\/span>)<\/span>;<\/span>
\n J<\/span>(<\/span>0<\/span>,<\/span>0<\/span>)<\/span> =<\/span> –<\/span>20.0<\/span>*<\/span>x[<\/span>0<\/span>]<\/span>;<\/span> J<\/span>(<\/span>0<\/span>,<\/span>1<\/span>)<\/span> =<\/span> 10.0<\/span>;<\/span>
\n J<\/span>(<\/span>1<\/span>,<\/span>0<\/span>)<\/span> =<\/span> –<\/span>1.0<\/span>;<\/span> J<\/span>(<\/span>1<\/span>,<\/span>1<\/span>)<\/span> =<\/span> 0.0<\/span>;<\/span>
\n return<\/span> J;<\/span>
\n }<\/span>;<\/span><\/p>\n

Lsqnonlin::<\/span>Options opts;<\/span>
\n opts.<\/span>verbose =<\/span> true<\/span>;<\/span><\/p>\n

Lsqnonlin solver<\/span>(<\/span>fun,<\/span> jac)<\/span>;<\/span>
\n Vec x0 =<\/span> {<\/span> –<\/span>1.2<\/span>,<\/span> 1.0<\/span> }<\/span>;<\/span>
\n auto<\/span> res =<\/span> solver.<\/span>solve<\/span>(<\/span>x0,<\/span> opts)<\/span>;<\/span><\/p>\n

std::<\/span>printf<\/span>(<\/span>"\\\\n ———- \u6c42\u89e3\u7ed3\u675f ———-\\\\n"<\/span>)<\/span>;<\/span>
\n printVec<\/span>(<\/span>" x"<\/span>,<\/span> res.<\/span>x)<\/span>;<\/span>
\n printVec<\/span>(<\/span>" residual"<\/span>,<\/span> res.<\/span>residual)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" cost = %.6e\\\\n"<\/span>,<\/span> res.<\/span>cost)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" iterations = %d\\\\n"<\/span>,<\/span> res.<\/span>iterations)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" exitFlag = %d\\\\n\\\\n"<\/span>,<\/span> res.<\/span>exitFlag)<\/span>;<\/span>
\n }<\/span><\/p>\n

\/\/ ———- \u7528\u4f8b 2: \u6307\u6570\u62df\u5408（\u6570\u503c\u96c5\u53ef\u6bd4）———-<\/span>
\n {<\/span>
\n std::<\/span>printf<\/span>(<\/span>"———- \u7528\u4f8b 2: \u6307\u6570\u62df\u5408 y = a*exp(b*t) ———-\\\\n"<\/span>)<\/span>;<\/span>
\n const<\/span> std::<\/span>vector<<\/span>double<\/span>><\/span> t =<\/span> {<\/span>0.0<\/span>,<\/span> 0.5<\/span>,<\/span> 1.0<\/span>,<\/span> 1.5<\/span>,<\/span> 2.0<\/span>,<\/span> 2.5<\/span>,<\/span> 3.0<\/span>}<\/span>;<\/span>
\n const<\/span> std::<\/span>vector<<\/span>double<\/span>><\/span> y =<\/span> {<\/span>2.60<\/span>,<\/span> 1.80<\/span>,<\/span> 1.30<\/span>,<\/span> 0.91<\/span>,<\/span> 0.63<\/span>,<\/span> 0.44<\/span>,<\/span> 0.31<\/span>}<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" \u771f\u503c a\u22482.5, b\u2248-0.7，\u521d\u503c p0 = [1.0, 0.0]\\\\n\\\\n"<\/span>)<\/span>;<\/span><\/p>\n

auto<\/span> fun =<\/span> [<\/span>&<\/span>]<\/span>(<\/span>const<\/span> Vec&<\/span> p)<\/span> -><\/span> Vec {<\/span>
\n Vec r<\/span>(<\/span>t.<\/span>size<\/span>(<\/span>)<\/span>)<\/span>;<\/span>
\n for<\/span> (<\/span>size_t i =<\/span> 0<\/span>;<\/span> i <<\/span> t.<\/span>size<\/span>(<\/span>)<\/span>;<\/span> ++<\/span>i)<\/span>
\n r[<\/span>i]<\/span> =<\/span> p[<\/span>0<\/span>]<\/span>*<\/span>std::<\/span>exp<\/span>(<\/span>p[<\/span>1<\/span>]<\/span>*<\/span>t[<\/span>i]<\/span>)<\/span> –<\/span> y[<\/span>i]<\/span>;<\/span>
\n return<\/span> r;<\/span>
\n }<\/span>;<\/span><\/p>\n

Lsqnonlin::<\/span>Options opts;<\/span>
\n opts.<\/span>verbose =<\/span> true<\/span>;<\/span>
\n Lsqnonlin solver<\/span>(<\/span>fun)<\/span>;<\/span> \/\/ \u4f7f\u7528\u6570\u503c\u96c5\u53ef\u6bd4<\/span>
\n Vec p0 =<\/span> {<\/span> 1.0<\/span>,<\/span> 0.0<\/span> }<\/span>;<\/span>
\n auto<\/span> res =<\/span> solver.<\/span>solve<\/span>(<\/span>p0,<\/span> opts)<\/span>;<\/span><\/p>\n

std::<\/span>printf<\/span>(<\/span>"\\\\n ———- \u6c42\u89e3\u7ed3\u675f ———-\\\\n"<\/span>)<\/span>;<\/span>
\n printVec<\/span>(<\/span>" p (a, b)"<\/span>,<\/span> res.<\/span>x)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" cost = %.6e\\\\n"<\/span>,<\/span> res.<\/span>cost)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" iterations = %d\\\\n"<\/span>,<\/span> res.<\/span>iterations)<\/span>;<\/span>
\n std::<\/span>printf<\/span>(<\/span>" exitFlag = %d\\\\n\\\\n"<\/span>,<\/span> res.<\/span>exitFlag)<\/span>;<\/span>
\n }<\/span>
\n return<\/span> 0<\/span>;<\/span>
\n}<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

\/\/
\n\/\/ lsqnonlin.cpp
\n\/\/ MATLAB \u9ed8\u8ba4\u7b97\u6cd5 trust-region-reflective \u7684\u72ec\u7acb\u5b9e\u73b0
\n\/\/ \u7f16\u8bd1: g -O2 -stdc17 lsqnonlin_trr.cpp -o lsqtrr
\n\/\/
\n#include
\n#include
\n#include
\n#include
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