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EPSEB - EGT ALGEBRA PRACTICE 3 LINEAR SYSTEMS 9 / 11 / 10
- Enter the packet LinearAlgebra with(LinearAlgebra):
- Go to the Maple "Help" and look for how to generate a random matrix or a random vector, that is, a matrix or a vector with elements generated randomly. Then, generate a square random matrix A of range 4 and a random vector b of range also 4.
LinearAlgebra[RandomMatrix] - construct a random Matrix
LinearAlgebra[RandomVector] - construct a random Vector
Calling Sequence
RandomMatrix(r, c, den, gen, outopts) RandomVector[o](d, den, gen, outopts)
Parameters
r - (optional) non-negative integer; row dimension of the resulting Matrix c - (optional) non-negative integer; column dimension of the resulting Matrix d - (optional) non-negative integer; dimension of the resulting Vector den - (optional) equation of the form density = p where p is a numeric value with 0 <= p <= 1; probability that a generated value is assigned to an entry in the resulting Matrix or Vector gen - (optional) equation of the form generator = f where f is a generator of values for the entries [o] - (optional) use either [row] or [column] to specify the orientation of the resulting Vector outopts - (optional) equation of the form outputoptions=list; constructor options for the result object
A:=RandomMatrix(4); b:=RandomVector(4);
A :=
25 50 K 50 K 38
94 10 K 22 K 18
12 K 16 45 87
K 2 K 9 K 81 33
b :=
K 50
K 80
- Solve the linear system AX = b by means LinearSolve s:=LinearSolve(A,b);
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s :=
K^3360923
K 148060
K^11994517
K^1131999
- Look in the Maple help for how to perform Gaussian elimination of a matrix, do it with the matrix (A|b), and solve the linear system AX = b transformed by the Gaussian elimination.
LinearAlgebra[Generic][GaussianElimination] - perform
Gaussian elimination on a Matrix
LinearAlgebra[Generic][ReducedRowEchelonForm] - compute
the reduced row echelon form of a Matrix
LinearAlgebra[Generic][RREF] - compute the reduced row
echelon form of a Matrix
Calling Sequence
GaussianEliminationF GaussianEliminationF ReducedRowEchelonFormF ReducedRowEchelonFormF ReducedRowEchelonFormF ReducedRowEchelonFormF RREFF RREFF RREFF RREFF
Parameters
F - the domain of computation, a field
A - rectangular Matrix over values in F
r - name
d - name
B:=; C:=GaussianElimination(B); s:=LinearSolve(B);
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OOOOOOOO p,L,U:=LUDecomposition(A); d:=LinearSolve(L,b); s:=LinearSolve(U,d);
p, L, U :=
K 2
K^1140
25 50 K 50 K 38
0 K 178 166 3122
d :=
K^4164
K^127828
K^1131999
s :=
K^3360923
K 148060
K^11994517
K^1131999
- Generate a random matrix A of 5 rows and 3 columns and a random vector b of range 5 and solve the over-determined system AX = b solving the normal system ATAX = ATb
A:=RandomMatrix(5,3); b:=RandomVector(5); At:=Transpose(A): N:=At.A; T:=At.b; LinearSolve(N,T);
A :=
K 32 72 K 62
K 1 42 K 33
52 18 K 68
K 13 K 59 K 67
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b :=
K 82
K 70
N :=
5097 17477 K 2137
192 K 2137 14190
T :=
K 10115
K 6128
K^180354946502
K^124340954901
- Look in the Maple help for how to perform QR decomposition of a matrix, do it with the matrix A, and solve the over-determined system AX = b using the QR decomposition
LinearAlgebra[QRDecomposition] - compute QR factorization
of a Matrix
Calling Sequence
QRDecomposition(A, fs, out, c, outopts, ...)
Parameters
A - Matrix fs - (optional) BooleanOpt(fullspan); selects whether a full QR factorization is performed out - (optional) equation of the form output = obj where obj is one of 'Q', R', 'NAG', or 'rank', or a list containing one or more of these names; selects result objects to compute c - (optional) BooleanOpt(conjugate); selects whether conjugations are performed in the Gram-Schmidt process outopts - (optional) equation of the form outputoptions[o] = list where o is one of 'Q', 'R', or 'NAG', and list contains options to pass to the result constructor; constructor options for the result object ... - (optional) additional instances of outputoptions[o] = list
Q,R:=QRDecomposition(A); Qt:=Transpose(Q): Qtb:=Qt.b: LinearSolve(R,Qtb);
