Finding Roots of Vector SystemsDoing one Vector Newton StepEigenanalysisLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==Let's consider roots of the following systemQyQ+SSJmRzYiLCgqJkkieEdGJSIiI0kieUdGJSIiJiIiIiomRihGK0YqRikhIiItSSRleHBHRiU2IywmKiZGKEYsRipGLEYsRi5GLEYsRiw=QyQ+SSJnRzYiLCoqJkkieEdGJSIiIkkieUdGJSIiJEYpKiZGKCIiI0YqRi1GKSomRihGK0YqRilGKSEiJEYpRik=These can be found using the fsolve command, giving input values close to the desired root.QyQ+SSZyb290MUc2Ii1JJ2Zzb2x2ZUdGJTYkPCQvSSJmR0YlIiIhL0kiZ0dGJUYsPCQvSSJ4R0YlIiIiL0kieUdGJUYyRjI=QyY+SSZ4c3Rhckc2Ii1JJHJoc0clKnByb3RlY3RlZEc2IyZJJnJvb3QxR0YlNiMiIiJGLT5JJnlzdGFyR0YlLUYnNiMmRis2IyIiI0YtQyQ+SSZyb290Mkc2Ii1JJ2Zzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkPCQvSSJmR0YlIiIhL0kiZ0dGJUYvPCQvSSJ4R0YlISIiL0kieUdGJUY1IiIiSo 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=QyQ+SSRqMTJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZkdGJUkieUdGJTwkL0kieEdGJSIiIi9GMUY1RjU=QyQ+SSRqMjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZ0dGJUkieEdGJTwkL0YxIiIiL0kieUdGJUY0RjQ=QyQ+SSRqMjJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJWV2YWxHRig2JC1JJWRpZmZHRig2JEkiZ0dGJUkieUdGJTwkL0kieEdGJSIiIi9GMUY1RjU=Then we could set up the linear equations that the next iterate x1 and y1 satisfy QyQ+SSRlcTFHNiIvLCgqJkkkajExR0YlIiIiLCZJInhHRiVGKiEiIkYqRipGKiomSSRqMTJHRiVGKiwmSSJ5R0YlRipGLUYqRipGKi1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGNDYkSSJmR0YlPCQvRixGKi9GMUYqRioiIiFGKg==QyQ+SSRlcTJHNiIvLCgqJkkkajIxR0YlIiIiLCZJInhHRiVGKiEiIkYqRipGKiomSSRqMjJHRiVGKiwmSSJ5R0YlRipGLUYqRipGKi1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUklZXZhbEdGNDYkSSJmR0YlPCQvRixGKi9GMUYqRioiIiFGKg==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. LUkmc29sdmVHNiI2JDwkSSRlcTFHRiRJJGVxMkdGJDwkSSJ4R0YkSSJ5R0YkIn one final topic, we show here how to enter a matrix and find its eigenvalues and eigenvectorsQyQ+SSNNMUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjNyQ3JCIiIiIiIzckRi8iIiRGLg==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 LUktRWlnZW52ZWN0b3JzRzYiNiNJI00xR0YkJSFHTTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMTUxODRYLCUpYW55dGhpbmdHNiJGJVtnbCEiJSEhISMlIiMiIyIiIiIiI0YnIiIkRiU=TTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMjE2NjRYKiUqYWxnZWJyYWljRzYiRiVbZ2whIyUhISEiIyIjLCYiIiMiIiIqJCIiJiNGKEYnRigsJkYnRihGKSEiIkYlTTdSMApJN1JUQUJMRV9TQVZFLzQzMDczMjE3ODRYLCUqYWxnZWJyYWljRzYiRiVbZ2whIiUhISEjJSIjIiMsJCokLCYqJCIiJiMiIiIiIiNGLEYsRiwhIiJGLUYsLCQqJCwmRixGLEYpRi5GLkYtRixGJQ==