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chapter8-9

# chapter8-9 - Linear Systems and Gauss Elimination Silvana...

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Linear Systems and Gauss Elimination Silvana Ilie MTH510 – Numerical Analysis Department of Mathematics, Ryerson University Linear SystemsandGauss Elimination – p. 1/17

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Chapter Objectives Understanding how to implement forward elimination and back substitution as in Gauss elimination. Understanding how to count flops to evaluate the efficiency of an algorithm. Understanding the concepts of singularity and ill-condition. Understanding how partial pivoting is implemented and how it differs from complete pivoting. Recognizing how the banded structure of a tridiagonal system can be exploited to obtain extremely efficient solutions. Linear SystemsandGauss Elimination – p. 2/17
Example: Electrical circuits Problem: determine currents and voltages in resistor circuits Kirchhoff’s Current Law: algebraic sum of all currents entering a node is zero: i = 0 Kirchhoff’s Voltage Law: algebraic sum of potential differences in any loop must be zero: ξ - iR = 0 ξ – electromotive force of voltage sources R – resistance of a resistor Ohm’s Law: voltage drop across an ideal resistor = current × resistance: Linear SystemsandGauss Elimination – p. 3/17

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Example: Electrical circuits (cont’d) i 65 2 1 3 4 5 6 i 43 i 32 i 54 i 52 i 12 Kirchhoff’s Current Law: i 12 + i 52 + i 32 = 0 i 65 - i 52 - i 54 = 0 i 43 - i 32 = 0 i 54 - i 43 = 0 Linear SystemsandGauss Elimination – p. 4/17
Example: Electrical circuits (cont’d) V 6 = 0 V R = 10Ω R = 10Ω R = 5Ω R = 15Ω R = 5Ω R = 20Ω 2 1 3 4 5 6 V 1 = 200 V i 65 2 1 3 4 5 6 i 43 i

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chapter8-9 - Linear Systems and Gauss Elimination Silvana...

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