Method quarters, c=[
0.25 , 0.5, 0.75, 1]
AUBV (scaled)
| 0.225 | 0 | -0 | -0 |
| -0.826024 | 0.225 | -0 | -0 |
| -1.5043 | -0.0909628 | 0.225 | -0 |
| 0.627021 | 1.14274 | -0.470412 | 0.225 |
|
| 1 | 0.025 | -0.05 | -0.0265625 |
| 1 | 1.10102 | 0.438012 | 0.11113 |
| 1 | 2.12026 | 1.06811 | 0.392465 |
| 1 | -0.524352 | -0.200635 | 0.144197 |
|
| 0.627021 | 1.14274 | -0.470412 | 0.225 |
| 0 | 0 | 0 | 1 |
| -0.801265 | -1.19016 | 0.496108 | 0.930233 |
| -0.853065 | -1.00014 | 0.434625 | 0.387597 |
|
| 1 | -0.524352 | -0.200635 | 0.144197 |
| 0 | 0 | 0 | 0 |
| 0 | 0.565087 | -0.0138326 | 0.414979 |
| 0 | 1.03098 | -0.000461087 | 0.0138326 |
|
Maximum coefficient scaled is 2.120260 .
irks(0.225,[0.25,0.5,0.75,1], [0.25,0.575],[ 1 0; 0.1 1; ])
[alpha,beta,gamma,stageerror]
| 0.000542057 | 0.0166649 | -0.0149371 | -0.00921875 |
| 0 | -0 | 0 | 0.0654692 |
| 0.25 | -0.191414 | 0.0476644 | 0.174301 |
| 0.575 | -0.192213 | 0.111588 | 0.000542057 |
Error estimation
phi=[-64.000000 192.000000 -192.000000 64.000000 ]
phi1=[239.717738 -263.576607 111.717738 -11.929435 ] phi0=-75.929435 gives y^{p+1}(t_n-theta*h) at theta=0.671599
M(infty)
| 0 | 0 | -0 | -0 |
| 24.4184 | 3.82924 | -0.879274 | -0.451728 |
| 59.7488 | 8.83768 | -2.32665 | -0.871599 |
| 93.1529 | 15.3723 | -4.0704 | -1.50259 |
Method thirds, c=[
0.333333 , 0.666667, 1, 1]
AUBV (scaled)
| 0.225 | 0 | 0 | 0 |
| 0.044517 | 0.225 | 0 | 0 |
| -2.01786 | 0.830968 | 0.225 | 0 |
| 1.94301 | 0.829634 | -0.633485 | 0.225 |
|
| 1 | 0.108333 | -0.0388889 | -0.037963 |
| 1 | 0.39715 | 0.114766 | -0.0185427 |
| 1 | 1.96189 | 0.787283 | -0.110337 |
| 1 | -1.36416 | -0.584549 | 0.471606 |
|
| 1.94301 | 0.829634 | -0.633485 | 0.225 |
| -0 | 0 | 0 | 1 |
| -2.0431 | 0.416663 | 0.147848 | 0.70922 |
| -0.709484 | -0.894444 | 0.462238 | 0.378251 |
|
| 1 | -1.36416 | -0.584549 | 0.471606 |
| 0 | 0 | 0 | 0 |
| 0 | 0.769368 | 0.0923798 | 0.554279 |
| 0 | 0.763439 | -0.0153966 | -0.0923798 |
|
Maximum coefficient scaled is 2.043099 .
irks(0.225,[0.333333,0.666667,1,1], [0.3,0.4],[ 1 0; -0.5 1; ])
[alpha,beta,gamma,stageerror]
| 0.000542057 | 0.0840728 | -0.082345 | -0.00923868 |
| 0 | -0 | 0 | 0.0128233 |
| 0.3 | -0.240648 | 0.120648 | 0.0863238 |
| 0.4 | -0.0481946 | 0.188195 | 0.000542057 |
Error estimation
phi=[0.000000 0.000000 0.000000 0.000000 ]
phi1=[81.000000 -81.000000 20.661594 6.338406 ] phi0=-27.000000 gives y^{p+1}(t_n-theta*h) at theta=0.500000
M(infty)
| -0 | -0 | -0 | 0 |
| -40.584 | -2.96589 | 1.6422 | 0.232887 |
| -51.583 | -4.12388 | 1.85087 | 0.496815 |
| -140.409 | -10.1942 | 4.85813 | 1.11502 |
Method lambda_1_quarter, c=[
0.333333 , 0.666667, 1, 1]
AUBV (scaled)
| 0.25 | 0 | -0 | -0 |
| -0.406574 | 0.25 | 0 | -0 |
| 0.0141676 | -0.211928 | 0.25 | 0 |
| 0.265232 | 0.958121 | -0.39668 | 0.25 |
|
| 1 | 0.0833333 | -0.0555556 | -0.0462963 |
| 1 | 0.82324 | 0.38216 | 0.0984875 |
| 1 | 0.947761 | 0.773126 | 0.527848 |
| 1 | -0.0766731 | -0.160956 | 0.0741344 |
|
| 0.265232 | 0.958121 | -0.39668 | 0.25 |
| 0 | 0 | 0 | 1 |
| -0.432251 | -0.911675 | 0.388324 | 0.892308 |
| -0.673028 | -0.726297 | 0.342506 | 0.369231 |
|
| 1 | -0.0766731 | -0.160956 | 0.0741344 |
| 0 | 0 | 0 | 0 |
| 0 | 0.0632947 | -0.0575285 | 0.517756 |
| 0 | 0.687588 | -0.00639206 | 0.0575285 |
|
Maximum coefficient scaled is 0.958121 .
irks(0.25,[0.333333,0.666667,1,1], [0.25,0.555556],[ 1 0; 0.333333 1; ])
[alpha,beta,gamma,stageerror]
| 0.00390625 | 0.0198682 | -0.0141391 | -0.0122599 |
| -0 | 0 | 0 | 0.0552837 |
| 0.25 | -0.198305 | 0.0594159 | 0.155894 |
| 0.555556 | -0.203447 | 0.144805 | 0.00390625 |
Error estimation
phi=[0.000000 0.000000 0.000000 0.000000 ]
phi1=[81.000000 -81.000000 35.302711 -8.302711 ] phi0=-27.000000 gives y^{p+1}(t_n-theta*h) at theta=0.500000
M(infty)
| 0 | 0 | -0 | -0 |
| 20.3871 | 4.21432 | -0.807728 | -0.604842 |
| 49.1489 | 9.47382 | -2.12095 | -1.19015 |
| 81.0421 | 17.0874 | -3.92125 | -2.09338 |