We store all s nonzero entries consecutively in a one dimensional array val[] so that val[j] stores the jth nonzero in the order from left to right, and top to bottom. We also maintain two extra auxiliary arrays to provide access to the individual matrix entries. Specifically we maintain one integer array col[] of size s such that col[j] is the column in which the jth nonzero appears.
This compares favorably with the 8N 2 bytes required with the 2D array representation. A Krylov-subspace method when A is symmetric positive definite. Is there ever a reason to explicitly compute the inverse of a matrix? Yes, if it's asked for on an exam. In practice, it's almost never necessary.
It's twice as fast and has better numerical accuracy and stability properties. Exercises Add a method frobenius to Matrix that returns the Frobenius norm of the matrix. The Frobenius norm is the square root of sum of the squares of all the entries. Add a method normInfinity that returns the infinity norm of the matrix.
The infinity norm a. Add a method trace that returns the trace of the matrix. The trace is the sum of the diagonal entries. Add a method isSymmetric that returns true if the matrix is symmetric and false otherwise. Add a method isTridiagonal that returns true if the matrix is tridiagonal , and false otherwise. Given an N-by-N array a[][] , write a code fragment to transpose a in-place. That is, use at most a few extra variables of storage. Add a method plusEquals that takes a Matrix B as input and over-writes the invoking matrix with the sum of itself and B.
Suppose you run Gaussian elimination without exchanging rows. Examine the residual error and observe that the solution vector has no significant digits of accuracy in many coordinates. The matrix A is well-conditioned, so it is not a result of the problem being ill-conditioned as in XYZ.
Instead, it is because partial pivoting is unstable, and this input highlights this defect. Name your program PartialPivotStability. Solve the following upper triangular linear system of equations by hand using back-substitution. Solve the following linear system of equations by hand using Gaussian elimination and back-substitution. You should obtain the upper triangular system in the previous exercise along the way.
Solve the following linear system of equations. Area of a triangle and ccw. Formulas for the area of a triangle have been known for years. In the 17th century Descartes and Fermat used linear algebra to gain new insight into geometric problems. For example, the following determinant gives twice the signed area of the triangle with vertices a, b, and c. This ccw test is a useful primitive for convex hull and other computational geometry algorithms.
Generalizes naturally to higher dimensions for tetrahedron and other simplices. In-circle test. Determine if the point d lies inside or outside the circle defined by the three points a, b, and c in the plane. Application: primitive operation in Delaunay triangulation algorithms. Assuming that a, b, c are labeled in counterclockwise order around the circle, the following determinant is positive if d is inside the circle, negative if d is outside the circle, and zero if all four points are cocircular.
This generalizes naturally to higher dimensions, e. Find the eigenvalues and eigenvectors of the following matrix. Suppose you are presented with the problem of computing c T A -1 d. Explain how to do it without explicitly computing A Now the desired answer is c T x. Whenever you see an inverse in a formula, always think of it as solving an equation rather than computing an inverse.
Creative Exercises Complex matrices. Create an abstract data type to represent complex matrices. Lights out. Implement a solver for the games Lights Out.
Feasibility detection. Can't find a non zero pivot and the current right hand side is nonzero. Certificate of infeasibility.
Tridiagonal matrices. Implement a data type TridiagonalMatrix that implements a tridiagonal matrix using three 1-D arrays. Your algorithm should run in linear time. Strassen's algorithm. Compare vs. Gaussian elimination.
Special matrices. Create classes DiagonalMatrix , TridiagonalMatrix using inheritance and override the methods for solving a linear system of equations and matrix multiplication Markov chains. Markov chains are a simple mathematical tool for modeling behavioral patterns. Widely used in many scientific areas including queuing theory, statistics, modeling population processes, and gene prediction.
Glass and Hall distinguished 7 states in their social mobility study: Professional, high administrative Managerial Inspectional, supervisory, non-manual high grade Non-manual low grade Skilled manual Semi-skilled manual Unskilled manual The table below presents the data from their study. Entry i, j is the probability of transitioning from state i to j. Hilbert matrix. Write a program Hilbert.
All Hilbert matrices are invertible. Answer : Jama matrix inverter reports that the matrix is not invertible. The Hilbert matrix is ill-conditioned, and most linear algebra packages have difficulty inverting this matrix for sufficiently large N. Note that although it inverts smaller matrices without reporting an error, there is substantial error in the results.
If Markov chain M is irreducible possible to get from any state to any other state after finitely many transitions and aperiodic, then it it called ergodic. Detailed balance. Hint : sum over j. Random walk on a ring. Suppose you have a circle of N nodes and you start at node 1. At each step, you flip a fair coin and go clockwise or counterclockwise accordingly. Calculate the probability that each vertex other than 1 is the last one visited. Solution : all have equal likelihood of being last!
Leontief input-output model. Branch of economics that uses linear algebra to model interdependencies of industries. Wassily Leontief won Nobel Prize in Economics description. Leontief divided American economy into 81 sectors petroleum, textiles, transportation, chemicals, steel, agriculture, etc.
For example, producing 1 unit of petroleum requires 0. Units measured in millions of dollars. Petroleum 0. Multiply Ax to get b [, , , Leontief closed model. Total production of each sector equals total consumption.
Leontief open model. Coal 0. Leslie matrix model. In population ecology, the Leslie matrix models the age distribution of a population this is naturally segmented into age classes. The Leslie matrix L is constructed by: The follow table lists the birth and survival probabilities for female New Zealand sheep. The original source is: [from G. Age years Birth rate Survival rate 0. In this example 1. Use eigenvalues to analyze.
Stationary distribution. Maximum cardinality matching. Given a bipartite G graph with N vertices on each side, find a matching of maximum cardinality. It contains exercises, many of these being frequently asked interview questions. A Solutions Manual including detailed solutions to every exercise was published concurrently. Top 3 QuantNet bestselling book in , , , and All the books purchased from FE Press are signed and personalized by the authors. Sample Sections: 1.
Here are key links:. Links to websites for each semester at MIT: web. My friend Pavel Grinfeld at Drexel has sent me a collection of interesting problems -- mostly elementary but each one with a small twist. These are part of his larger teaching site called LEM. This video standard describes a system for encoding and decoding a "Codec" that engineers have defined for applications like High Definition TV. It is not expected that you will know the meaning of every word -- your book author does not know either.
The point is to see an important example of a "standard" that is created by an industry after years of development so all companies will know what coding system their products must be consistent with.
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