backslash

Description

Backslash is the left matrix division: X=A\B is a solution to A*X=B.

If A is square and non-singular X=A\B is equivalent to X=inv(A)*B in exact arithmetic, but the computations are more accurate and cheaper in floating point arithmetic. Hence, to compute the solution of the linear system of equations A*X=B, the backslash operator should be used, and the inv function should be avoided.

In the case where A is square, the solution X can be computed either from LU factorization or from a linear least squares solver. If the condition number of A is smaller than 1/(10*%eps) (i.e. if A is well conditioned), the LU factorization with row pivoting is used. If not (i.e. if A is poorly conditioned), then X is the minimum-norm solution which minimizes ||A*X-B|| using a complete orthogonal factorization of A (i.e. X is the solution of a linear least squares problem).

If A is not square, X is a least square solution, i.e. norm(A*X-B) is minimal (Euclidean norm). If A is full column rank, the least square solution, X=A\B, is uniquely defined (there is a unique X which minimizes norm(A*X-B)). If A is not full column rank, then the least square solution is not unique, and X=A\B, in general, is not the solution with minimum norm (the minimum norm solution is X=pinv(A)*B).

A.\B is the matrix with (i,j) entry A(i,j)\B(i,j). If A (or B) is a scalar A.\B is equivalent to A*ones(B).\B (or A.\(B*ones(A)).

A\.B is an operator with no predefined meaning. It may be used to define a new operator (see overloading) with the same precedence as * or /.

Examples

A=[
   9.   -36.    30.
  -36.   192.  -180.
   30.  -180.   180.
];
b=[
   3.
  -24.
   30.
];
x=A\b
A*x-b // close to zero

A=rand(3,2);
b=[1;1;1];
x=A\b;
y=pinv(A)*b;
x-y
A=rand(2,3);b=[1;1];
x=A\b;
y=pinv(A)*b;
x-y, A*x-b, A*y-b

// if rank is deficient
A=rand(3,1)*rand(1,2);
b=[1;1;1];
x=A\b;
y=pinv(A)*b;
A*x-b, A*y-b
A=rand(2,1)*rand(1,3);
b=[1;1];
x=A\b;
y=pinv(A)*b;
A*x-b, A*y-b

// A benchmark of several linear solvers

[A,descr,ref,mtype] = ReadHBSparse(SCI+..
   "/modules/umfpack/demos/bcsstk24.rsa");

b = zeros(size(A,1),1);

tic();
res = umfpack(A,'\',b);
mprintf('\nTime with umfpack: %.3f\n',toc());

tic();
res = linsolve(A,b);
mprintf('\ntime with linsolve: %.3f\n',toc());

tic();
res = A\b;
mprintf('\ntime with backslash: %.3f\n',toc());

See also

• slash — (/) right divisions. System's feed back. Comments
• inv — matrix inverse
• pinv — pseudoinverse
• linsolve — linear equation solver
• umfpack — solve sparse linear system
• kron .\. — Kronecker left and right divisions
• overloading — display, functions and operators overloading capabilities

History