Numerical Methods

Linear systems graphicallySingular matricesSingular matrices do not have a unique solutionParallel planes indicate that the system has no solutionOverlapping planes result in infinite solutionsMatrices with multiple solutions or no solutionsStatistically these are rare (if you generate random matrices)no solutioninfinite solutionsIll-conditioned systemsPlanes that almost overlap are almost singular are ill conditionedSmall change in one or more coefficients results in large changes in the solutionThis results in large changes due to small errorsnoise, roundoff

ill conditionedIll-conditioned systemsDeterminantsCalculating determinantsLU DecompositionLU Decomposition (continued)LU Decomposition (continued)LowermatrixUppermatrixfactorsusedfor GELU Decomposition (examples)LU with Partial PivotingFeatures of LU DecompositionMatrix InversionComputing the Matrix InverseMatrix Inversion (example)Matrix and vector normsTypes of matrix normsEuclidean norm:

1-norm:

sum the magnitude of all values in each column, taking the largest value as the norm

uniform matrix norm:

sum the magnitude of all values in each row, taking the largest value as the normMatrix Norms (example)Properties of matrix normsSystem conditioningIll-conditioned matrices are highly sensitiveSmall changes in input create large changes in outputSmall changes include noise and roundoff!Ill-conditioned systems are difficult to solve computationallyBest you can really do is be aware of them

ill conditionedMatrix condition numberCondition number derivedConditioning and loss of precisionMatrix condition number (cont.)Condition number (examples)Condition number (examples)