Wyrażenia matematyczne

Podczas laboratorium zaimplementujemy kilka klas pozwalających na reprezentację prostych wyrażeń matematycznych (funkcji jednej lub kilku zmiennych) jako grafów. Podobna reprezentacja jest rdzeniem funkcji oferowanych na przykład przez https://live.sympy.org/ lub https://www.derivative-calculator.net/. Nie będziemy jednak “parsowali” wyrażeń, czyli budowali grafowej reprezentacji na podstawie specyfikacji tekstowej. Zamiast tego, wyrażenia będą budowane za pomocą odpowiednio skonstruowanego kodu.

Założona funkcjonalność:

budowa wyrażeń
obliczanie wartości wyrażeń/funkcji dla różnych wartości zmiennych
wydruk wyrażeń
symboliczne obliczanie pochodnych [zadanie domowe]

Jakie klasy będą potrzebne

Przeanalizujmy wyrażenie $ 2x^3 + x^2 -2x + 7 $

Jego elementami są:

stałe (np 2 i 7)
zmienne (x)
funkcja potęgowa ($x^3$, $x^2$) – Power(arg,p)
operacja mnożenia Prod(args…)
operacja dodawania Sum(args…)

Wyrażenie $ 2x^3 + x^2 -2x + 7 $ może być zapisane jako Sum(Prod(2,Power(x,3)),Power(x,2),Prod(-2,x),7). Zakładając, że każdy z tych elementów będzie obiektem, otrzymamy reprezentację w postaci acyklicznego grafu, jak na poniższym rysunku. Węzłami grafu są stałe, zmienne, operacje i funkcje, natomiast krawędzie odpowiadają argumentom poszczególnych operacji.

W zasadzie jest to drzewo, którego liśćmi są zmienne i stałe. Ponieważ zmienne mogą powtarzać się, liście będące zmiennymi zostały złączone

Zaimplementujemy więc klasy, jak na poniższym diagramie i dodamy wspólnego przodka, abstrakcyjną klasę Node. Niebieskie linie to asocjacje/agregacje łączące węzły nadrzędne (opreacje, funkcje) i ich argumenty.

Argumentem Power jest jakieś wyrażenie, czyli Node.
Sum może mieć więcej argumentów, stąd agregacja elementów Node
Prod (product) to iloczyn większej liczby czynników Node

Implementujemy klasy

Node

abstract public class Node {
    int sign=1;
    Node minus(){
        sign = -1;
        return this;
    }
    Node plus(){
        sign = 1;
        return this;
    }
    int getSign(){return sign;}
 
    /**
     * Oblicza wartość wyrażenia dla danych wartości zmiennych
     * występujących w wyrażeniu
     */
    abstract double evaluate();
 
    /**
     *
     * zwraca tekstową reprezentację wyrażenia
     */
    public String toString(){return "";}
 
    /**
     * 
     * Zwraca liczbę argumentów węzła
     */
    int getArgumentsCount(){return 0;}
 
}

W klasie Node wprowadzono atrybut sign znak, dziedziczony przez klasy potomne. Potraktujmy to następująco. Załóżmy, że reprezentacją tekstową węzła jest abcd

jeśli sign<0 reprezentacją tekstową będzie -(abcd) [w nawiasach]
jeśli sign>=0 będzie to po prostu abcd [bez nawiasów]

Constant

public class Constant extends Node {
    double value;
    Constant(double value){
        this.sign = value<0?-1:1;
        this.value = value<0?-value:value;
    }
 
 
    @Override
    double evaluate() {
        return sign*value;
    }
 
    @Override
    public String toString() {
        String sgn=sign<0?"-":"";
//        return sgn+Double.toString(value);
        DecimalFormat format = new DecimalFormat("0.#####",new DecimalFormatSymbols(Locale.US));
        return sgn+format.format(value);
    }
}

Stosując DecimalFormat pozbędziemy się niepotrzebnych zer na końcu wartości double.

Variable

Klasa Variable reprezentuje zmienną, która może być użyta w wyrażeniach.

public class Variable extends Node {
    String name;
    Double value;
    Variable(String name){
        this.name = name;
    }
    void setValue(double d){
        value = d;
        // value = d<0?-value:value;
        // sign = d<0?-1:1;
    }
 
 
    @Override
    double evaluate() {
         //return sign*value;
         return value;
    }
 
 
    @Override
    public String toString() {
        String sgn=sign<0?"-":"";
        return sgn+name;
    }
 
}

Power

Power jest jedyną zaimplementowaną funkcją. Zauważmy, że $Power(x,-1)=\frac{1}{x}$. Argumentem Power nie musi być pojedyncza zmienna. Może to być np.: Power(Sum(x,2),-2), czyli $\frac{1}{(x+2)^2}$

W kodzie pokazano przykładową implementację toString() z próbą wstawienia nawiasów w takich przypadkach, jak $-(-x^5)$ lub $(x+1)^2$

public class Power extends Node {
    double p;
    Node arg;
    Power(Node n,double p){
        arg = n;
        this.p = p;
    }
 
    @Override
    double evaluate() {
        double argVal = arg.evaluate();
        return Math.pow(argVal,p);
    }
 
 
    int getArgumentsCount(){return 1;}
 
 
    @Override
    public String toString() {
        StringBuilder b = new StringBuilder();
        if(sign<0)b.append("-");
        int argSign = arg.getSign();
        int cnt = arg.getArgumentsCount();
        boolean useBracket = false;
        if(argSign<0 ||cnt>1)useBracket = true;
        String argString = arg.toString();
        if(useBracket)b.append("(");
        b.append(argString);
        if(useBracket)b.append(")");
        b.append("^");
        b.append(p);
        return b.toString();
    }
 
}

Sum

Klasa Sum jest wyrażeniem reprezentującą sumę argumentów. Zawiera listę węzłów List<Node> args = new ArrayList<>();

Teoretycznie, lista argumentów może liczyć 0,1,2,… elementów. Metoda add() dodaje składniki do listy

public class Sum extends Node {
 
    List<Node> args = new ArrayList<>();
 
    Sum(){}
 
    Sum(Node n1, Node n2){
        args.add(n1);
        args.add(n2);
    }
 
 
    Sum add(Node n){
        args.add(n);
        return this;
    }
 
    Sum add(double c){
        args.add(new Constant(c));
        return this;
    }
 
    Sum add(double c, Node n) {
        Node mul = new Prod(c,n);
        args.add(mul);
        return this;
    }
 
    @Override
    double evaluate() {
        double result =0;
        oblicz sumę wartości zwróconych przez wywołanie evaluate skłądników sumy
        return sign*result;
    }
 
    int getArgumentsCount(){return args.size();}
 
    public String toString(){
        StringBuilder b =  new StringBuilder();
        if(sign<0)b.append("-(");
 
        //zaimplementuj
 
        if(sign<0)b.append(")");
        return b.toString();
    }
 
 
}

Prod

Prod to iloczyn (ang. product). Reprezentuje wyrażenie będące iloczynem czynników. Metoda mul() dodaje czynnik do listy.

public class Prod extends Node {
    List<Node> args = new ArrayList<>();
 
    Prod(){}
 
    Prod(Node n1){
        args.add(n1);
    }
    Prod(double c){
        wywołaj konstruktor jednoargumentowy przekazując new Constant(c)
    }
 
    Prod(Node n1, Node n2){
        args.add(n1);
        args.add(n2);
    }
    Prod(double c, Node n){
        wywołaj konstruktor dwuargumentowy
    }
 
 
 
    Prod mul(Node n){
        args.add(n);
        return this;
    }
 
    Prod mul(double c){
        ???
    }
 
 
    @Override
    double evaluate() {
        double result =1;
        // oblicz iloczyn czynników wołąjąc ich metodę evaluate
        return sign*result;
    }
    int getArgumentsCount(){return args.size();}
 
 
    public String toString(){
        StringBuilder b =  new StringBuilder();
        if(sign<0)b.append("-");
        // ...
        zaimplementuj
        return b.toString();
    }
 
 
}

Testujemy implementację

Test 1

    static void buildAndPrint(){
        Variable x = new Variable("x");
        Node exp = new Sum()
                .add(2.1,new Power(x,3))
                .add(new Power(x,2))
                .add(-2,x)
                .add(7);
        System.out.println(exp.toString());
 
    }

Oczekiwany wynik: 2.1*x^3 + x^2 + (-2)*x + 7

Test 2

    static void buildAndEvaluate(){
        Variable x = new Variable("x");
        Node exp = new Sum()
                .add(new Power(x,3))
                .add(-2,new Power(x,2))
                .add(-1,x)
                .add(2);
        for(double v=-5;v<5;v+=0.1){
            x.setValue(v);
            System.out.printf(Locale.US,"f(%f)=%f\n",v,exp.evaluate());
        }

Sprawdź, jakie są pierwiastki? Możesz użyć

https://en.wikipedia.org/wiki/Bisection_method lub
https://en.wikipedia.org/wiki/Newton%27s_method po dodaniu obliczania pochodnych.

Test 3

    static void defineCircle(){
        Variable x = new Variable("x");
        Variable y = new Variable("y");
        Node circle = new Sum()
                .add(new Power(x,2))
                .add(new Power(y,2))
                .add(8,x)
                .add(4,y)
                .add(16);
        System.out.println(circle.toString());
 
        double xv = 100*(Math.random()-.5);
        double yv = 100*(Math.random()-.5);
        x.setValue(xv);
        y.setValue(yv);
        double fv = circle.evaluate();
        System.out.print(String.format("Punkt (%f,%f) leży %s koła %s",xv,yv,(fv<0?"wewnątrz":"na zewnątrz"),circle.toString()));
}

Znajdź i wypisz 100 punktów leżących wewnątrz okręgu…. Oczywiście napisz kod, który je znajduje

Symboliczne obliczanie pochodnych

Zadanie do realizacji w domu.

Pochodna wyrażenia jest również wyrażeniem… Dodajmy metodę diff() w klasie Node

    abstract Node diff(Variable var);

Dodajmy jeszcze metodę

    abstract boolean isZero();

która będzie służyła do optymalizacji wywołań (pomijamy wyrażenia o wartości 0).

Pochodne

Constant

Zawsze pochodną stałej jest zero.

    @Override
    Node diff(Variable var) {
        return new Constant(0);
    }

Variable

Pochodna $\frac{d}{dx}(x)=1$, natomiast $\frac{d}{dx}(y)=0$

    Node diff(Variable var) {
        if(var.name.equals(name))return new Constant(sign);
        else return new Constant(0);
    }

Power

Z definicji $\frac{d}{dx}(x^n)=nx^{n-1}$, ale tak naprawdę powinniśmy potraktować to jako złożenie funkcji $Power(arg(x)),p$, stąd

$$\frac{d}{dx}(arg(x)^n) = n\cdot arg(x)^{n-1}\cdot \frac{d}{dx}arg(x)$$

    Node diff(Variable var) {
        Prod r =  new Prod(sign*p,new Power(arg,p-1));
        r.mul(arg.diff(var));
        return r;
    }

Sum

Pochodna sumy jest sumą pochodnych. Najprostsza implementacja jest podana poniżej.

    Node diffVanilla(Variable var) {
        Sum r = new Sum();
        for(Node n:args){
            r.add(n.diff(var));
        }
        return r;
    }

Niestety, dużo pochodnych składowych będzie zerowych. Stąd propozycja usprawnienia przez sprawdzenie przed dodaniem, czy wyrażenie nie jest zerem.

Prod

Ogólny algorytm iloczynu jest podany np. tu https://en.wikipedia.org/wiki/Product_rule. Zauważmy, że stosuje się to także do funkcji stałych (w projekcie Constant):

$$\frac{d}{dx}(2\cdot f(x))=\frac{d}{dx}(2)\cdot f(x) + 2\cdot \frac{d}{dx}f(x)$$ $$=0\cdot f(x) + 2\cdot \frac{d}{dx} f(x)$$

Stąd najprostsza implementacja ma postać:

    Node diffVanilla(Variable var) {
        Sum r = new Sum();
        for(int i=0;i<args.size();i++){
            Prod m= new Prod();
            for(int j=0;j<args.size();j++){
                Node f = args.get(j);
                if(j==i)m.mul(f.diff(var));
                else m.mul(f);
            }
            r.add(m);
        }
        return r;
    }

Niestety, w jej wyniku może powstać sporo wyrażeń zerowych…

Testy

Test 1

static void diffPoly() {
        Variable x = new Variable("x");
        Node exp = new Sum()
                .add(2,new Power(x,3))
                .add(new Power(x,2))
                .add(-2,x)
                .add(7);
        System.out.print("exp=");
        System.out.println(exp.toString());
 
        Node d = exp.diff(x);
        System.out.print("d(exp)/dx=");
        System.out.println(d.toString());
 
    }

Wynik

exp=2*x^3 + x^2 + (-2)*x + 7
d(exp)/dx=0*x^3 + 2*3*x^2*1 + 2*x^1*1 + 0*x + (-2)*1 + 0

Test 2

    static void diffCircle() {
        Variable x = new Variable("x");
        Variable y = new Variable("y");
        Node circle = new Sum()
                .add(new Power(x,2))
                .add(new Power(y,2))
                .add(8,x)
                .add(4,y)
                .add(16);
        System.out.print("f(x,y)=");
        System.out.println(circle.toString());
 
        Node dx = circle.diff(x);
        System.out.print("d f(x,y)/dx=");
        System.out.println(dx.toString());
        System.out.print("d f(x,y)/dy=");
        Node dy = circle.diff(y);
        System.out.println(dy.toString());
 
    }

Wynik

f(x,y)=x^2 + y^2 + 8*x + 4*y + 16
d f(x,y)/dx=2*x^1*1 + 2*y^1*0 + 0*x + 8*1 + 0*y + 4*0 + 0
d f(x,y)/dy=2*x^1*0 + 2*y^1*1 + 0*x + 8*0 + 0*y + 4*1 + 0

Dodaj rozszerzenia i optymalizacje

Pomijaj przy sumowaniu wyrażenia zerowe Przed utworzeniem węzłów Sum i Prod umieść wyrażenia na listach.

Jeżeli lista jest pusta, zwróć new Constant(0)
Jeżeli lista zawiera jeden element, po prostu zwróć go

Wynik dla circle:

f(x,y)=x^2 + y^2 + 8*x + 4*y + 16
d f(x,y)/dx=2*x^1*1 + 8*1
d f(x,y)/dy=2*y^1*1 + 4*1

Dodaj metodę simplify() , która zastąpi iloczyn stałych pojedynczą stałą. Spłaszcz strukturę Constant()*Product(args) do pojedynczego iloczynu.

exp=2*x^3 + x^2 + (-2)*x + 7
d(exp)/dx=6*x^2 + 2*x^1 + (-2)
f(x,y)=x^2 + y^2 + 8*x + 4*y + 16
d f(x,y)/dx=2*x^1 + 8
d f(x,y)/dy=2*y^1 + 4

Dodaj klasy odpowiadające innym funkcjom, np. exp(), log(), sin(), cos(), itd… A w przyszłości może uda się rozwinąć program do postaci podobnej do https://www.derivative-calculator.net/

expand (opcjonalne)

Dla zainteresowanych

Zaimplementuj metodę expand() na podstawie https://en.wikipedia.org/wiki/Multinomial_theorem

Na pewno interesującym problemem jest tu generacja wszystkich krotek $(k_1,k_2,\dots,k_m)$ spełniających $k_1+\dots+k_m=n$

Implementacja metody gradientu prostego

Dla zainteresowanych. Nie jest to zadanie do wykonania, ale gotowy kod do poeksperymentowania…

O metodzie możesz przeczytać tu https://pl.wikipedia.org/wiki/Metoda_gradientu_prostego.

Jest to podstawowa metoda stosowana w uczeniu maszynowym. Zmiennymi są wagi modelu, w kolejnych iteracjach są one uaktualniane, aby zminimalizować funkcję celu.

W przypadku wielu klasycznych algorytmów funkcja celu jest funkcją wypukłą i metoda zmierza do minimum globalnego. Dla sieci neuronowych z reguły funkcja ma wiele minimów lokalnych i proces optymalizacji może zmierzać do jednego z nich.

Kluczowym elementem współczesnych implementacji jest symboliczne wyznaczenie pochodnych w postaci grafu obliczeniowego, który może być wykonywany na GPU.

W poniższym przykładzie minimalizowana jest funkcja celu $$f(x_0,\dots,x_{n-1})=\sum_{i=0}^{n-1}a_i x_i^2$$ Jej współczynniki $a_i$ oraz początkowe wartości zmiennych $x_i$ są losowane, stąd za każdym razem przebieg będzie inny. Dla konkretnej funkcji celu mozna/należy eksperymentalnie dobrać liczbę iteracji maxIter oraz stałą uczenia learningRate.

    static void testOptimization(){
        // rozmiar problemu
        int n=10;
        // hiperparametry
        int maxIter=1000;
        double learningRate=0.1;
 
        Random rand = new Random();
        // Utwórz zmienne
        List<Variable> vars= new ArrayList<>();
        for(int i=0;i<n;i++){
            vars.add(new Variable("var"+i));
        }
 
        // Zdefiniuj funkcję celu jako sum(a_i*x_i^2)
        Sum goal = new Sum();
        for(int i=0;i<n;i++){
            double a = rand.nextDouble();
            Node b = new Power(vars.get(i),2);
            goal.add(new Prod(new Constant(a),b));
        }
 
        // Wyznacz symbolicznie gradienty funkcji celu względem zmiennych x_i
        ArrayList<Node> grads = new ArrayList<>();
        for(int i=0;i<vars.size();i++){
            grads.add(goal.diff(vars.get(i)));
        }
 
        // Wylosuj poczatkowe wartosci zmiennych z przedziału [-500,500]
        for(int i=0;i<vars.size();i++){
            vars.get(i).setValue(rand.nextDouble()*1000-500);
        }
 
        // iteracja
        double[]grad_values = new double[n];
        for(int iter=0;iter<maxIter;iter++){
 
            // oblicz wartosc funkcji celu
            double goal_value = goal.evaluate();
            System.out.printf(Locale.US,"[%d] goal:%f\n",iter,goal_value);
 
            // oblicz wartosci gradientów
            // symboliczne gradienty korzystają z tych samych zmiennych, co funkcja celu
            for(int i=0;i<n;i++){
                grad_values[i] = grads.get(i).evaluate();
            }
 
            // uaktualnij wartosci zmiennych wg. wzoru x_i = x_i-lr*gradient_i
            for(int i=0;i<n;i++){
                Variable v = vars.get(i);
                v.setValue(v.getValue()-learningRate*grad_values[i]);
            }
        }
        double goal_value = goal.evaluate();
        System.out.printf(Locale.US,"[at end] goal:%f",goal_value);
    }

Poniżej przykładowy wynik (bardzo udany, często są znacznie gorsze). Oczywiście rozwiązaniem jest wektor z samymi zerami. Jeśli chcesz możesz wydrukować wartości rozwiązania w kolejnych iteracjach.

[0] goal:497945.411139
[1] goal:358421.084154
[2] goal:261261.945738
[3] goal:193012.852980
[4] goal:144625.847295
[5] goal:109985.516384
[6] goal:84935.043961
[7] goal:66631.223694
[8] goal:53116.025550
[9] goal:43031.027181
[10] goal:35426.331320
[11] goal:29632.149331
[12] goal:25172.079663
[13] goal:21704.227883
[14] goal:18980.992937
[15] goal:16821.424551
[16] goal:15092.089356
[17] goal:13693.727905
[18] goal:12551.876689
[19] goal:11610.222737
[20] goal:10825.854697
[21] goal:10165.839926
[22] goal:9604.736015
[23] goal:9122.766116
[24] goal:8704.469798
[25] goal:8337.697383
[26] goal:8012.854503
[27] goal:7722.330436
[28] goal:7460.062523
[29] goal:7221.202133
[30] goal:7001.856962
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[818] goal:0.000408
[819] goal:0.000400
[820] goal:0.000391
[821] goal:0.000383
[822] goal:0.000376
[823] goal:0.000368
[824] goal:0.000360
[825] goal:0.000353
[826] goal:0.000345
[827] goal:0.000338
[828] goal:0.000331
[829] goal:0.000325
[830] goal:0.000318
[831] goal:0.000311
[832] goal:0.000305
[833] goal:0.000299
[834] goal:0.000292
[835] goal:0.000286
[836] goal:0.000281
[837] goal:0.000275
[838] goal:0.000269
[839] goal:0.000264
[840] goal:0.000258
[841] goal:0.000253
[842] goal:0.000248
[843] goal:0.000242
[844] goal:0.000237
[845] goal:0.000233
[846] goal:0.000228
[847] goal:0.000223
[848] goal:0.000218
[849] goal:0.000214
[850] goal:0.000210
[851] goal:0.000205
[852] goal:0.000201
[853] goal:0.000197
[854] goal:0.000193
[855] goal:0.000189
[856] goal:0.000185
[857] goal:0.000181
[858] goal:0.000177
[859] goal:0.000174
[860] goal:0.000170
[861] goal:0.000167
[862] goal:0.000163
[863] goal:0.000160
[864] goal:0.000157
[865] goal:0.000153
[866] goal:0.000150
[867] goal:0.000147
[868] goal:0.000144
[869] goal:0.000141
[870] goal:0.000138
[871] goal:0.000135
[872] goal:0.000133
[873] goal:0.000130
[874] goal:0.000127
[875] goal:0.000124
[876] goal:0.000122
[877] goal:0.000119
[878] goal:0.000117
[879] goal:0.000115
[880] goal:0.000112
[881] goal:0.000110
[882] goal:0.000108
[883] goal:0.000105
[884] goal:0.000103
[885] goal:0.000101
[886] goal:0.000099
[887] goal:0.000097
[888] goal:0.000095
[889] goal:0.000093
[890] goal:0.000091
[891] goal:0.000089
[892] goal:0.000087
[893] goal:0.000086
[894] goal:0.000084
[895] goal:0.000082
[896] goal:0.000080
[897] goal:0.000079
[898] goal:0.000077
[899] goal:0.000076
[900] goal:0.000074
[901] goal:0.000072
[902] goal:0.000071
[903] goal:0.000069
[904] goal:0.000068
[905] goal:0.000067
[906] goal:0.000065
[907] goal:0.000064
[908] goal:0.000063
[909] goal:0.000061
[910] goal:0.000060
[911] goal:0.000059
[912] goal:0.000058
[913] goal:0.000056
[914] goal:0.000055
[915] goal:0.000054
[916] goal:0.000053
[917] goal:0.000052
[918] goal:0.000051
[919] goal:0.000050
[920] goal:0.000049
[921] goal:0.000048
[922] goal:0.000047
[923] goal:0.000046
[924] goal:0.000045
[925] goal:0.000044
[926] goal:0.000043
[927] goal:0.000042
[928] goal:0.000041
[929] goal:0.000040
[930] goal:0.000040
[931] goal:0.000039
[932] goal:0.000038
[933] goal:0.000037
[934] goal:0.000036
[935] goal:0.000036
[936] goal:0.000035
[937] goal:0.000034
[938] goal:0.000034
[939] goal:0.000033
[940] goal:0.000032
[941] goal:0.000031
[942] goal:0.000031
[943] goal:0.000030
[944] goal:0.000030
[945] goal:0.000029
[946] goal:0.000028
[947] goal:0.000028
[948] goal:0.000027
[949] goal:0.000027
[950] goal:0.000026
[951] goal:0.000026
[952] goal:0.000025
[953] goal:0.000025
[954] goal:0.000024
[955] goal:0.000024
[956] goal:0.000023
[957] goal:0.000023
[958] goal:0.000022
[959] goal:0.000022
[960] goal:0.000021
[961] goal:0.000021
[962] goal:0.000020
[963] goal:0.000020
[964] goal:0.000020
[965] goal:0.000019
[966] goal:0.000019
[967] goal:0.000018
[968] goal:0.000018
[969] goal:0.000018
[970] goal:0.000017
[971] goal:0.000017
[972] goal:0.000017
[973] goal:0.000016
[974] goal:0.000016
[975] goal:0.000016
[976] goal:0.000015
[977] goal:0.000015
[978] goal:0.000015
[979] goal:0.000014
[980] goal:0.000014
[981] goal:0.000014
[982] goal:0.000013
[983] goal:0.000013
[984] goal:0.000013
[985] goal:0.000013
[986] goal:0.000012
[987] goal:0.000012
[988] goal:0.000012
[989] goal:0.000012
[990] goal:0.000011
[991] goal:0.000011
[992] goal:0.000011
[993] goal:0.000011
[994] goal:0.000010
[995] goal:0.000010
[996] goal:0.000010
[997] goal:0.000010
[998] goal:0.000010
[999] goal:0.000009
[at end] goal:0.000009

Table of Contents

Wyrażenia matematyczne

Jakie klasy będą potrzebne

Implementujemy klasy

Node

Constant

Variable

Power

Sum

Prod

Testujemy implementację

Test 1

Test 2

Test 3

Symboliczne obliczanie pochodnych

Pochodne

Constant

Variable

Power

Sum

Prod

Testy

Test 1

Test 2

Dodaj rozszerzenia i optymalizacje

expand (opcjonalne)

Implementacja metody gradientu prostego