x

x =

   1.999999998137355

history

history =

                   0   3.000000000000000
   1.500000000000000   3.000000000000000
   1.500000000000000   2.250000000000000
   1.875000000000000   2.250000000000000
   1.875000000000000   2.062500000000000
   1.968750000000000   2.062500000000000
   1.968750000000000   2.015625000000000
   1.992187500000000   2.015625000000000
   1.992187500000000   2.003906250000000
   1.998046875000000   2.003906250000000
   1.998046875000000   2.000976562500000
   1.999511718750000   2.000976562500000
   1.999511718750000   2.000244140625000
   1.999877929687500   2.000244140625000
   1.999877929687500   2.000061035156250
   1.999969482421875   2.000061035156250
   1.999969482421875   2.000015258789062
   1.999992370605469   2.000015258789062
   1.999992370605469   2.000003814697266
   1.999998092651367   2.000003814697266
   1.999998092651367   2.000000953674316
   1.999999523162842   2.000000953674316
   1.999999523162842   2.000000238418579
   1.999999880790710   2.000000238418579
   1.999999880790710   2.000000059604645
   1.999999970197678   2.000000059604645
   1.999999970197678   2.000000014901161
   1.999999992549419   2.000000014901161
   1.999999992549419   2.000000003725290

a=history(:,1)

a =

                   0
   1.500000000000000
   1.500000000000000
   1.875000000000000
   1.875000000000000
   1.968750000000000
   1.968750000000000
   1.992187500000000
   1.992187500000000
   1.998046875000000
   1.998046875000000
   1.999511718750000
   1.999511718750000
   1.999877929687500
   1.999877929687500
   1.999969482421875
   1.999969482421875
   1.999992370605469
   1.999992370605469
   1.999998092651367
   1.999998092651367
   1.999999523162842
   1.999999523162842
   1.999999880790710
   1.999999880790710
   1.999999970197678
   1.999999970197678
   1.999999992549419
   1.999999992549419

b=history(:,2);
m=(a+b)/2;
err = 2 - m

err =

   0.500000000000000
  -0.250000000000000
   0.125000000000000
  -0.062500000000000
   0.031250000000000
  -0.015625000000000
   0.007812500000000
  -0.003906250000000
   0.001953125000000
  -0.000976562500000
   0.000488281250000
  -0.000244140625000
   0.000122070312500
  -0.000061035156250
   0.000030517578125
  -0.000015258789062
   0.000007629394531
  -0.000003814697266
   0.000001907348633
  -0.000000953674316
   0.000000476837158
  -0.000000238418579
   0.000000119209290
  -0.000000059604645
   0.000000029802322
  -0.000000014901161
   0.000000007450581
  -0.000000003725290
   0.000000001862645

plot(m)
plot(m,'*')
plot(err,'*r')
plot(log(abs(err)),'r*')
polyfit(1:length(err),log(abs(err)),1)

ans =

  -0.693147180559945   0.000000000000000

coeffs=polyfit(1:length(err),log(abs(err)),1)

coeffs =

  -0.693147180559945   0.000000000000000

m=coeffs(1); B=coeffs(2)

B =

     4.232268566237738e-16

k=1:length(err);
plot(k,log(abs(err)),'r*',k,m*k+B,'g')
plot(k,log(abs(err)),'r*',k,m*k+B,'b')
m

m =

  -0.693147180559945

exp(m)

ans =

   0.500000000000000

log(0.5)

ans =

  -0.693147180559945

err

err =

   0.500000000000000
  -0.250000000000000
   0.125000000000000
  -0.062500000000000
   0.031250000000000
  -0.015625000000000
   0.007812500000000
  -0.003906250000000
   0.001953125000000
  -0.000976562500000
   0.000488281250000
  -0.000244140625000
   0.000122070312500
  -0.000061035156250
   0.000030517578125
  -0.000015258789062
   0.000007629394531
  -0.000003814697266
   0.000001907348633
  -0.000000953674316
   0.000000476837158
  -0.000000238418579
   0.000000119209290
  -0.000000059604645
   0.000000029802322
  -0.000000014901161
   0.000000007450581
  -0.000000003725290
   0.000000001862645

diary