The map

In[1]:=

f[{x_, y_}] = {x^3 - 2 x^2 y + y, y^3 - 2 x^2}

Out[1]=

{x^3 + y - 2 x^2 y, -2 x^2 + y^3}

The derivative

In[2]:=

Df[{x_, y_}] = {{3 x^2 - 4 x y, -2 x^2 + 1}, {-4 x, 3 y^2}} ; MatrixForm[Df[{x, y}]]

Out[3]//MatrixForm=

( 2 2 ) 3 x - 4 x y 1 - 2 x 2 -4 x 3 y

The linear approximation about p evaluated at q

In[4]:=

L[p_, q_] := Simplify[f[p] + Df[p] . (q - p)]

The initial point, it's image, and a check to see that Df is invertible at this point

In[14]:=

guess = {1, 1} f[guess] Det[Df[guess]]

Out[14]=

{1, 1}

Out[15]=

{0, -1}

Out[16]=

-7

Finding a better guess

In[17]:=

L[guess, {x, y}] guess = {x, y} /. First[Solve[L[guess, {x, y}] == {.1, -.8}, {x, y}]] f[guess]

Out[17]=

{2 - x - y, -4 x + 3 y}

Out[18]=

{0.9285714285714286`, 0.9714285714285714`}

Out[19]=

{0.09686588921282802`, -0.8077784256559767`}

Finding a better guess

In[20]:=

L[guess, {x, y}] guess = {x, y} /. First[Solve[L[guess, {x, y}] == {.1, -.8}, {x, y}]] f[guess]

Out[20]=

{1.7491253644314873` - 1.0214285714285718` x - 0.7244897959183674` y, -0.10893294460641378` - 3.7142857142857144` x + 2.8310204081632655` y}

Out[21]=

{0.9259726396783751`, 0.9707665420067657`}

Out[22]=

{0.09999929346363357`, -0.8000122304158221`}

Finding a better guess

In[23]:=

L[guess, {x, y}] guess = {x, y} /. First[Solve[L[guess, {x, y}] == {.1, -.8}, {x, y}]] f[guess]

Out[23]=

{1.7415344970862643` - 1.0233370413549987` x - 0.7148506588658756` y, -0.11482619803423111` - 3.7038905587135003` x + 2.8271630372393206` y}

Out[24]=

{0.9259707012795794`, 0.9707683285322534`}

Out[25]=

{0.10000000001596931`, -0.7999999999982196`}

Converted by Mathematica  (February 27, 2004)