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 Ratings
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| Reviewed: |
Jan 16, 2004 by Mathematics |
| Overview: |
A Java applet for numerically solving systems of linear equations. Systems of up to 15 equations can be handled, and approximately 20 significant figures are given in the solutions (though not all of these may be correct, as noted below).
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| Learning Goals: |
With careful guidance, students could use this ability to make and verify conjectures about the solvability or non-solvability of systems depending on the coefficient matrix, or to compute inverses of matrices. |
| Target Student Population: |
Linear Algebra students or anyone who needs to solve linear systems. |
| Prerequisite Knowledge or Skills: |
Basic knowledge of systems of linear equations. |
| Type of Material: |
Simulation |
| Recommended Uses: |
The only direct use is to solve systems of linear equations. With careful guidance, students could use this ability to make and verify conjectures about the solvability or non-solvability of systems depending on the coefficient matrix, or to compute inverses of matrices. |
| Technical Requirements: |
It requires a "Java-enabled" browser. (For one reviewer, the applet did not appear in Netscape 7.1 running on Mac OSX 10.1.) |
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| Strengths: |
The applet is straight-forward and intuitive: the user enters the number of equations in the system, and then clicks on an ?Enter Equations? button. This brings up a new window in which the coefficients and right-hand side of the system may be entered; clicking on a ?Done? button produces the solutions. The user may choose to enter an arbitrary coefficient matrix by hand, or to use a Hilbert matrix; the latter may allow for some interesting classroom experimentation.
The applet code is available at the site. |
| Concerns: |
One unfortunate feature is that the window containing the coefficients of the system is closed when the solutions are calculated. If one wants to solve the same system with a different right-hand side, for example, all coefficients have to be re-entered from scratch. This would make experimentation and conjecture rather tedious (unless small matrices, or the Hilbert matrix option, were used).
While answers are given to 20 significant figures, not all of these may be accurate. In fact, when the Hilbert matrix option is used, the last 5 or 6 digits are usually incorrect. Of course, the Hilbert matrix is known to be computationally hard, and comparison with results from Maple seem to show that all but 2 digits may be trusted for simpler matrices. It would have been useful to have some discussion of accuracy on the site.
The largest system that can be handled by the applet is 15 by 15, but this is not mentioned anywhere. Examination of the applet code, however, shows that if the user enters a number greater than 15 for the number of equations, the applet will use 15 instead. (This can be verified by asking for, say, 20 equations.) |
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Potential Effectiveness as a Teaching Tool |
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| Strengths: |
The applet efficiently carries out a computation which would be hard to do by hand. As mentioned above,
it could lend itself to experimentation and conjecture on the existence and behavior of solutions of linear systems, and possibly matrix inverses.
The applet can be used as an aid to a lecture or as a quick in-class demonstration. |
| Concerns: |
Classroom activities may be slowed down by the time needed to enter, or re-enter, the original matrix term-by-term. (This is probably unavoidable in matrix computations, though.) |
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Ease of Use for Both Students and Faculty |
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| Strengths: |
The applet is very easy to use. The average user can begin using the applet immediately. |
| Concerns: |
The ability to retain coefficients from one computation to another would greatly speed up certain types of calculation and experiment. |
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