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LU Decomposition

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LU Decomposition

Greg Beckham, Michael Sedivy

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

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;

}

…

}

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;

}

…

}

- 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

}

- Finds maximum

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;

}

- Performs row interchanges

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;

- 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?

- 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 |

- 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

- 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

Average Difference is 2.765471

Average Difference is 1.255924