7.9 Solving Systems of Equations

On this page:

7.9Solving Systems of Equations🔗 i

procedure
( matrix-solve MB[fail])→(U F(Matrix Number ))
M:(Matrix Number )
B:(Matrix Number )
fail:(-> F)=(λ ()(error ... ))

Returns the matrix X for which (matrix* MX) is B. M and B must have the same number of rows.

It is typical for B (and thus X) to be a column matrix, but not required. If B is not a column matrix, matrix-solve solves for all the columns in B simultaneously.

Examples:

> (define M(matrix [[75][3-2]]))
> (define B0(col-matrix [322]))
> (define B1(col-matrix [194]))
> (matrix-solve MB0)

- : (Array Real)

(array #[#[4] #[-5]])
> (matrix* M(col-matrix [4-5]))

- : (Array Integer)

(array #[#[3] #[22]])
> (matrix-solve MB1)

- : (Array Real)

(array #[#[2] #[1]])
> (matrix-cols (matrix-solve M(matrix-augment (list B0B1))))

- : (Listof (Array Real))

(list (array #[#[4] #[-5]]) (array #[#[2] #[1]]))

matrix-solve does not solve overconstrained or underconstrained systems, meaning that M must be invertible. If M is not invertible, the result of applying the failure thunk fail is returned.

matrix-solve is implemented using matrix-gauss-elim to preserve exactness in its output, with partial pivoting for greater numerical stability when M is not exact.

See vandermonde-matrix for an example that uses matrix-solve to compute Legendre polynomials.

procedure
( matrix-inverse M[fail])→(U F(Matrix Number ))
M:(Matrix Number )
fail:(-> F)=(λ ()(error ... ))

Wikipedia: Invertible Matrix Returns the inverse of M if it exists; otherwise returns the result of applying the failure thunk fail.

Examples:

> (matrix-inverse (identity-matrix 3))

- : (Array Real)

(array #[#[1 0 0] #[0 1 0] #[0 0 1]])
> (matrix-inverse (matrix [[75][3-2]]))

- : (Array Real)

(array #[#[2/29 5/29] #[3/29 -7/29]])
> (matrix-inverse (matrix [[12][1020]]))

matrix-inverse: contract violation

expected: matrix-invertible?

given: (array #[#[1 2] #[10 20]])
> (matrix-inverse (matrix [[12][1020]])(λ ()#f))

- : (U (Array Real) False)

#f

procedure
( matrix-invertible? M)→Boolean
M:(Matrix Number )

Returns #t when M is a square-matrix? and (matrix-determinant M) is nonzero.

procedure
( matrix-determinant M)→Number
M:(Matrix Number )

Wikipedia: Determinant Returns the determinant of M, which must be a square-matrix? .

Examples:

> (matrix-determinant (diagonal-matrix '(1234)))

- : Real

24
> (* 1234)

- : Integer [more precisely: Positive-Integer]

24
> (matrix-determinant (matrix [[12][1020]]))

- : Real

0
> (matrix-determinant (col-matrix [12]))

square-matrix-size: contract violation

expected: square-matrix?

given: (array #[#[1] #[2]])

top ← prev up next →