Finding Roots of Vector Systems Doing one Vector Newton Step Eigenanalysis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== Let's consider roots of the following system QyQ+SSJmRzYiLCgqJkkieEdGJSIiI0kieUdGJSIiJiIiIiomRihGK0YqRikhIiItSSRleHBHRiU2IywmKiZGKEYsRipGLEYsRi5GLEYsRiw= LCgqJkkieEc2IiIiI0kieUdGJSIiJiIiIiomRiRGKEYnRiYhIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmKiZGJEYpRidGKUYpRitGKUYp QyQ+SSJnRzYiLCoqJkkieEdGJSIiIkkieUdGJSIiJEYpKiZGKCIiI0YqRi1GKSomRihGK0YqRilGKSEiJEYpRik= LCoqJkkieEc2IiIiIkkieUdGJSIiJEYmKiZGJCIiI0YnRipGJiomRiRGKEYnRiZGJiEiJEYm These can be found using the fsolve command, giving input values close to the desired root. QyQ+SSZyb290MUc2Ii1JJ2Zzb2x2ZUdGJTYkPCQvSSJmR0YlIiIhL0kiZ0dGJUYsPCQvSSJ4R0YlIiIiL0kieUdGJUYyRjI= PCQvSSJ4RzYiJCIrMzVTcjYhIiovSSJ5R0YlJCIrTCQzLFEpISM1 QyY+SSZ4c3Rhckc2Ii1JJHJoc0clKnByb3RlY3RlZEc2IyZJJnJvb3QxR0YlNiMiIiJGLT5JJnlzdGFyR0YlLUYnNiMmRis2IyIiI0Yt JCIrMzVTcjYhIio= JCIrTCQzLFEpISM1 QyQ+SSZyb290Mkc2Ii1JJ2Zzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSJmR0YlIiIhL0kiZ0dGJUYvPCQvSSJ4R0YlISIiL0kieUdGJUY1IiIi PCQvSSJ4RzYiJCErTCQzLFEpISM1L0kieUdGJSQhKzM1U3I2ISIq So if we wanted numerically computed roots we could use this command in Maple. Let's investigate the steps that would go into a Newton's method to approximate roots near a given starting point. We will just do one step of Newton's method below, approximating the root found quite accurately above, starting at x=1, y=1. First we would find the entries of the Jacobian matrix at the current point QyQ+SSRqMTFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZkdGJUkieEdGJTwkL0YxIiIiL0kieUdGJUY0RjQ= JCEiIyIiIQ== QyQ+SSRqMTJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZkdGJUkieUdGJTwkL0kieEdGJSIiIi9GMUY1RjU= JCIiJSIiIQ== QyQ+SSRqMjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZ0dGJUkieEdGJTwkL0YxIiIiL0kieUdGJUY0RjQ= JCIiJyIiIQ== QyQ+SSRqMjJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZ0dGJUkieUdGJTwkL0kieEdGJSIiIi9GMUY1RjU= JCIiJyIiIQ== Then we could set up the linear equations that the next iterate x1 and y1 satisfy QyQ+SSRlcTFHNiIvLCgqJkkkajExR0YlIiIiLCZJInhHRiVGKiEiIkYqRipGKiomSSRqMTJHRiVGKiwmSSJ5R0YlRipGLUYqRipGKi1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGNDYkSSJmR0YlPCQvRixGKi9GMUYqRioiIiFGKg== LywoSSJ4RzYiJCEiIyIiISQhIiJGKCIiIkkieUdGJSQiIiVGKEYo QyQ+SSRlcTJHNiIvLCgqJkkkajIxR0YlIiIiLCZJInhHRiVGKiEiIkYqRipGKiomSSRqMjJHRiVGKiwmSSJ5R0YlRipGLUYqRipGKi1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGNDYkSSJmR0YlPCQvRixGKi9GMUYqRioiIiFGKg== LywoSSJ4RzYiJCIiJyIiISQhIzZGKCIiIkkieUdGJUYmRig= Then we could let Maple solve the linear system, giving output of the next approximation x1, y1 to the root. Note that the output is indeed closer to the root computed with fsolve above. LUkmc29sdmVHNiI2JDwkSSRlcTFHRiRJJGVxMkdGJDwkSSJ4R0YkSSJ5R0Yk PCQvSSJ4RzYiJCIrY2JiYjUhIiovSSJ5R0YlJCIreXh4eHghIzU= In one final topic, we show here how to enter a matrix and find its eigenvalues and eigenvectors QyQ+SSNNMUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjNyQ3JCIiIiIiIzckRi8iIiRGLg== LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKyU9OnRJJQ== QyQtSSV3aXRoRzYiNiNJLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUhIiI= Below, the eigenvalues are in the first column vector on the left. The corresponding eigenvectors are in the columns of the matrix on the left LUktRWlnZW52ZWN0b3JzRzYiNiNJI00xR0Yk NiQtJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkdGKTYjL0kkJWlkR0YpIitrO0syVi1JJ01hdHJpeEdGJjYjL0YuIisleUB0SSU= JSFH TTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMTUxODRYLCUpYW55dGhpbmdHNiJGJVtnbCEiJSEhISMlIiMiIyIiIiIiI0YnIiIkRiU=TTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMjE2NjRYKiUqYWxnZWJyYWljRzYiRiVbZ2whIyUhISEiIyIjLCYiIiMiIiIqJCIiJiNGKEYnRigsJkYnRihGKSEiIkYlTTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMjE3ODRYLCUqYWxnZWJyYWljRzYiRiVbZ2whIiUhISEjJSIjIiMsJCokLCYqJCIiJiMiIiIiIiNGLEYsRiwhIiJGLUYsLCQqJCwmRixGLEYpRi5GLkYtRixGJQ==