linear algebra - Calculating an inverse matrix in Matlab -

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i'm running optimization algorithm requires calculation of inverse of matrix. goal of algorithm eliminate negative values matrix , obtain new matrix b. basically, start known square matrices b , c of same size.

i start calculating matrix equal to:

a = b^-1 * c

or in matlab:

a = b\c;

i use because matlab told me b\c more accurate inv(b)*c.

the negative values in divided 2 , normalised it's rows have length of 1. using new a, calculate new b with:

(1/n) * * c' = b^-1

where n scaling factor (# of columns in a). new b used again in first step , these iterations continue until negatives in gone.

my problem have calculate b second equation , normalise it.

invb = (1/n)*a*c'; b = inv(invb);

i've been calculating b using inv(b^-1) after few iterations start getting messages b^-1 "close singular or badly scaled."

this algorithm works smaller matrices (around 70x70) when gets 500x500 start getting these messages.

are there better ways calculate inv(b^-1)?

you should head warnings singular matrices. results in numerical linear algebra tend break down move toward matrices high condition numbers. underlying idea if

a*b_1 = c

and we're solving problem (because using approximate numbers when use computers)

(a + matrix error)*b_2 = (c + vector error)

how close b_1 , b_2 function of matrix , vector errors? when has small condition number b_1 , b_2 close. when has large condition number b_1 , b_2 not close.

there informative piece of analysis on algorithm. @ each iteration, after you've found b, find use matlab find condition number of it. 

cond(b)

you see number climb rapidly. indicates every time iterate algorithm, should trust result b less , less.

problems crop time in numerical mathematics. if you'll working numerical algorithms should take time familiarize role of condition numbers in field , preconditioning techniques mentioned above. preferred text "numerical linear algebra" lloyd trefethen, text on numerical algebra should address of these issues.

best of luck, andrew


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