Well-Conditioned vs. Ill-Conditioned Systems of Equations
Explore the concepts of well-conditioned and ill-conditioned systems of equations in numerical methods. Learn how small changes in the coefficient matrix or right-hand side impact the solution vector. Delve into matrix determinants and the accuracy of solutions through practical examples.
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Autar Kaw Humberto Isaza the magnitude of the determinant is an indication of neither the condition of the matrix nor the accuracy of the solution" - Henry Thacher http://nm.MathForCollege.com Transforming Numerical Methods Education for STEM Undergraduates
? ?? ? ?? ? ? ? ??????, ? =max ??? ,1 < ? < ? ?=1 10 3 5 7 2.099 1 0 6 5 Find the row sum norm of the following matrix ? = ? ? = max ??? ,1 < ? < ? ?=1 = max 10 + 7 + 0 , 3 + 2.099 + 6 , 5 + 1 + 5 = max 10 + 7 + 0 , 3 + 2.099 + 6 , 5 + 1 + 5 = max 17,11.099,11 = 17.
Accuracy of Solution 20 15 10 45 x 1 . 1 = 3 . 2 249 7 751 x 2 5 1 3 9 x 3 Solve it on a computer using 6 significant digits with chopping . 0 9625 x 1 = . 1 05 x 2 . 0 999995 x 3
Accuracy of Solution 20 15 10 45 x 1 . 1 = 3 . 2 249 7 751 x 2 5 1 3 9 x 3 Solve it on a computer using 5 significant digits with chopping = . 0 3 x . 0 625 x 1 5 . 1 x 2 99995
What do you mean by ill-conditioned and well-conditioned system of equations? A system of equations is considered to be well-conditioned if a small change in the coefficient matrix or a small change in the right hand side results in a small change in the solution vector. A system of equations is considered to be ill-conditioned if a small change in the coefficient matrix or a small change in the right hand side results in a large change in the solution vector.
? ?=2 ? ?= 1 2 2 4 3.999 7.999 1 ? ?=4.001 ? ?= 3.999 1 2 2 4.000 3.999 7.998 ? ?= ? ?= 3.994 0.001388 4 1.001 2.001 2.001 3.998 7.999
? ?=2 ? ?=4 1 2 2 3 1 7 ? ?=1.999 ? ?=4.001 1 2 2 3 7.001 1.001 ? ?=4 ? ?=2.003 1.001 2.001 2.001 3.001 7 0.997
? ?= ? ?=2 1 2 2 4 ? = 7.999 ? = 2 3.999 7.999 1 ? ?=4.001 ? ?= 3.999 1 2 2 4.000 3.999 7.998 0.001 0.001 4 ? =4.001 = ? = 0.001 7.998 7.999 ? = 3.999 2 = 5.999 3.000 ? = 5.999 4.000 1 ? ? =0.001 7.999= 1.250 10 4 ? ? =5.999 = 2.9995 2
?/ ? ? / ? 2.9995 1.250 10 4 = = 23993
There is a relationship that exists between ? ? ? ? and and between ? ? ? ? and These relationships are ? ? ? ? ? ? ? ? 1 ? ? 1 < ? ? + ?
? ?=2 1 2 2 3.999 4 How many significant digits can I trust in the solution of the above system of equations? ? =1 2 ? 1= 3999 2000 1000 2 3.999 2000 ? = 5.999 ? 1 = 5999 ???? ? = ? ? 1 = 5.999 5999.4 = 35990
Assuming single precision with 23 bits used for the magnitude of mantissa for real numbers, the machine epsilon is ??? = 2 23= 0.119209 10 6 ????(?) ??? = 35990 0.119209 10 6= 0.4290 10 2 0.5 10 ? Comparing it with 0.4290 10 2 0.5 10 ? So two significant digits are at least correct in the solution vector as ? < 2
