Systems of linear equations
Systems of linear equations
• Ax=b
• A is non singular if
o A-1 exists
o A can be inverted
o |A| is not zero
o Az (0 if z (0
• Number of solutions
o If A is non singular - one and only one solution
o If A is nonsingular and b ( span(A) - infinite solutions,
o If A is nonsingular and b is not in the span(A) - no solutions
• Sensitivity and conditioning
o How sensitive is x to errors in b
o Reminder norms
▪ ||A||1=Max(j)(i|aij| - column
▪ ||A||(=Max(i)(j|aij| - rows
▪ ||AB||=1
o cond(I)=1
o cond(cA)=cond(A)
o cond(D)=(max|di|)/(min(|di|)
o cond(A) is not similar to determinant
o |cIn|=cn
• Upper diagonal matrix
o Simply solve one by one from the last
o Xi=(bi-(j=1,i-1aijxj)/aii
• Gauss method
o transform every matrix into a U matrix using a L matrix.
o Reminder left (pre) multiplication affects rows, right multiplication (post) affects columns
o Ax=b,MAz=Mb,z=(MA)-1Mb, z=A-1M-1Mb= A-1b=x
o APz=b,Ax=b,z=(AP)-1b=P-1x.
o If P is a permutation then P-1=PT and z=PTx
o Make consecutive left multiplication to transform any matrix into lan upper triangular (U) matrix. The resulting matrix is a lower triangular matrix (L) the resulting method is denoted the LU method
o MnMn-1…M1A=U,MA=U,A=M-1U=LU,L=A-1
o Algorithm
▪ Loop over rows i=1,n
• Loop over rows j=i+1,n
• aj*=aj*-(aij/aii)ai*
• End loop
▪ End loop
o Error level : ||r||/||A||||x|| ................
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