3 situations of solution existence (under the view of linear equations): 0,1 or infinite;
Computational complexity: n^3/3+...;
Gaussian-Jordan Method
Computational complexity: n^3/2+...;
Roundoff error
Form of floating number: f=± .d1 d2 ... dt * b^n (d1≠0);
Roundoff error: caused by the different magnitudes between the different columns;
Partial pivoting: search the position BELOW the pivotal position for the coefficient in maximum magnitude;
Complete pivoting: search the position BELOW and on the RIGHT of the pivotal position for the max coefficient;
Partial & Complete pivoting: whether using elementary column operation. The partial one is used more frequently because the elementary column operation is not easy to use;
the ill-conditioned system
the solution of an ill-conditioned system is extremely sensitive to a small perturbation on the coefficients;
Geometrical view: two linear systems are almost parallel so that their cross point will move sensitively when any one system moved;
How to notice the ill-condition of a linear system: enumerating ( it's not easy to find whether a system is ill-conditioned);
2 way to solve the problem: bite the bullet and compute the accurate solution, or redesign the experiment setup to avoid producing the ill-conditioned system. The latter one is better empirically. Finding a system is an ill-conditioned one as early as possible will save much time;
Row echelon form
Notation: E;
Cause: linear correlation between different column vectors and modified Gaussian elimination;
The echelon form (namely the position of pivots) is uniquely determined by the entries in A. However, the entries in E is not uniquely determined by A.
Basic column: the columns in A which contain the pivotal position;
Rank: the number of pivots = the number of nonzero rows in E = the number of basic columns in A;
Reduced row echelon form: produced by Gaussian-Jordan Method( [0 0 1 0]T ), notated by EA;
Both form and entries of EA is uniquely determined by A;
EA can show the hidden relationships among the different columns of A;
Consistency of linear system
A system is consistent if it has at least one solution. Otherwise, it is inconsistent.
When n (the number of equations) is two or three, the consistency of the system can be shown geometrically, the common point.
If n>3, we can judge through the following method:
In the augmented matrix [A|b], 0=a≠0 does not exist;
In [A|b], b is the nonbasic column;
rank(A|b)=rank(A);
b is the combination of the basic column in A.
Homogeneous system
Homogeneous and nonhomogeneous;
Trivial solution;
A homogeneous system must be a consistent system;
General solution: basic variable, free variable;
Nonhomogeneous system
General solution;
The system possesses a unique solution if and only if:
rank(A) = the number of the unknowns;
no free variable;
the associated homogeneous system only has a trivial solution;
Observation of LU: Advantages of LU factorization:
L:
a lower triangular matrix;
1's on the diagonal: means itself row plus other rows' multiplication with a scalar;
the entries below the diagonal record the multipliers used to eliminate;
U:
an upper triangular matrix;
the result of the elimination on A;
*L and U are unique;
proof: A=L1U1=L2U2 => L2'L1=U2U1', L2'L1 is a lower triangular matrix, U2U1' is an upper triangular matrix. They are equal to each other. So I=I => L2'L1=U2U1'=I.
*If exchanging of two rows is emerging during LU factorizing, the consistency of triangular form will be destroyed;
Advantages of LU factorization:
If only one system Ax=b need to be solved, the Gaussian Elimination is enough;
If more then one systems which coefficient matrices are the same need to be solved, the LU factorization is better;
Once the LU factors of A are known, any other system Ax=b can be solved in n^2 multiplications and n^2-n additions;
Existence of LU:
No zero pivot emerges during row reduction to upper triangular form with type III operation;
Another characterization method associated with principle submatrix: each leading principle submatrices is nonsingular;
