LU Decomposition

Greg Beckham, Michael Sedivy

Overview

Step 1

This is handled implicitly in the code by only calculating the diagonal for β

Step 2

Calculating βij

for(j = 0; j < n; j++) // This is the loop over columns of Crout's method {

for(i = 0; i < j; i++) // Equation (2.3.12) except for i = j { sum = a[i][j]; for(k = 0; k < i; k++) sum -= a[i][k] * a[k][j]; a[i][j] = sum; }

… }

Calculating αij

for(j = 0; j < n; j++) // This is the loop over columns of Crout's method{ … for(i = j; i < n; i++)// This is i=j of equation (2.3.12) and i=j+1 { // N-1 of equation (2.3.13) sum = a[i][j]; for(k = 0; k < j; k++) sum -= a[i][k] * a[k][j]; a[i][j] = sum; … } …

if(j != n - 1) // Divide by the pivot element { dum = 1.0/(a[j][j]); for(i = j + 1; i < n; i++) a[i][j] *= dum; } …}

Pivoting

• Initially finds largest element in each row• Used as a “scaling factor”, not sure of use other than

to rollover for(i = 0; i < n; i++) // Loop over the rows to get implicit scaling { // information big = 0.0; for(j = 0; j < n; j++) { if((temp = fabs(a[i][j])) > big) big = temp; }

if (big == 0.0) { printf("ERROR: Singular matrix\n"); }

// non-zero largest element. vv[i] = 1.0/big; // Save the scaling }

Pivoting

if((dum = vv[i] * fabs(sum)) >= big) { // Is the figure of merit for the pivot better than the best so far? big = dum; imax = i; }

• Finds maximum

Pivoting

if(j != imax) // Do we need to interchange rows? { for(k = 0; k < n; k++) // Interchange rows { dum = a[imax][k]; a[imax][k] = a[j][k]; a[j][k] = dum; }

d = -d; // change the parity of d vv[imax] = vv[j]; // interchange scale factor }

indx[j] = imax;

• Performs row interchanges

Related Questions• What is the advantage of LU(P) solver over GJ(P) solver?

(Complexity)– Both are O(N3)– After LU(P) is solved, more solutions supposed to be found in O (N2)

• Are you keeping L and U in the same matrix, or separate? Advantage/disadvantage?– LU are being created in place in the same matrix.– The advantage to this strategy is lower memory usage– The disadvantage is that the original matrix is lost

• I am somewhat confused with extraction of P in decomposition, and how it is then used in eq solving. Can you elaborate more?

Related Questions

• Cormen et al., p 824, used a single array instead of P. Needs careful explanation.– From Cormen et al. 825. “we dynamically

maintain the permutation matrix P as an array π, where π[i] = j means that the ith row of P contains a 1 in column j

|2| | 0 1 0 |π = |3| => P | 0 0 1 | |1| | 1 0 0 |

Related Questions

• How complex equations are solved? (in Text)– If only the right hand side vector is complex then

the operation can be performed by solving for the real part, then the imaginary

– If the matrix itself is complex then• Rewrite the algorithm for complex values• Split the real and imaginary parts into separate real

number and solve using existing algorithm– A * x – C * y = b– C * x + A * y = b

GJ vs. LUP: we found lup faster than gj, but…

GJ vs. LUP: lup not faster amortized

GJ vs. LUP

Average Difference is 2.765471

Average Difference is 1.255924