numpy.linalg.solve#

linalg.solve(a,b)[source]#

Solve a linear matrix equation, or system of linear scalar equations.

Computes the “exact” solution,x, of the well-determined, i.e., fullrank, linear matrix equationax = b.

Parameters:
a(…, M, M) array_like

Coefficient matrix.

b{(M,), (…, M, K)}, array_like

Ordinate or “dependent variable” values.

Returns:
x{(…, M,), (…, M, K)} ndarray

Solution to the system a x = b. Returned shape is (…, M) if b isshape (M,) and (…, M, K) if b is (…, M, K), where the “…” part isbroadcasted between a and b.

Raises:
LinAlgError

Ifa is singular or not square.

See also

scipy.linalg.solve

Similar function in SciPy.

Notes

Broadcasting rules apply, see thenumpy.linalg documentation fordetails.

The solutions are computed using LAPACK routine_gesv.

a must be square and of full-rank, i.e., all rows (or, equivalently,columns) must be linearly independent; if either is not true, uselstsq for the least-squares best “solution” of thesystem/equation.

Changed in version 2.0:The b array is only treated as a shape (M,) column vector if it isexactly 1-dimensional. In all other instances it is treated as a stackof (M, K) matrices. Previously b would be treated as a stack of (M,)vectors if b.ndim was equal to a.ndim - 1.

References

[1]

G. Strang,Linear Algebra and Its Applications, 2nd Ed., Orlando,FL, Academic Press, Inc., 1980, pg. 22.

Examples

Solve the system of equations:x0+2*x1=1 and3*x0+5*x1=2:

>>>importnumpyasnp>>>a=np.array([[1,2],[3,5]])>>>b=np.array([1,2])>>>x=np.linalg.solve(a,b)>>>xarray([-1.,  1.])

Check that the solution is correct:

>>>np.allclose(np.dot(a,x),b)True
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