7.9 Solving Systems of Equations

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

Returns the matrix X for which (matrix* M X) 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 [[7 5] [3 -2]]))
> (define B0 (col-matrix [3 22]))
> (define B1 (col-matrix [19 4]))
> (matrix-solve M B0)
eval:66:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: M7
  in: B0
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
eval:66:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: B08
  in: B0
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
Type Checker: Summary: 2 errors encountered
> (matrix* M (col-matrix [4 -5]))
eval:67:0: Type Checker: missing type for top-level
identifier;
either undefined or missing a type annotation
  identifier: M7
  in: -5
> (matrix-solve M B1)
eval:68:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: M7
  in: B1
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
eval:68:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: B19
  in: B1
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
Type Checker: Summary: 2 errors encountered
> (matrix-cols (matrix-solve M (matrix-augment (list B0 B1))))
eval:69:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: M7
  in: B1
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
eval:69:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: B08
  in: B1
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
eval:69:0: Type Checker: missing type for top-level identifier;
either undefined or missing a type annotation
  identifier: B19
  in: B1
  context...:
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:133:21: for-loop
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/utils/tc-utils.rkt:123:0: report-all-errors
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typecheck/tc-toplevel.rkt:515:0: tc-toplevel-form
   fail-to-succeed
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/tc-setup.rkt:39:0: tc-setup
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/typed-racket-lib/typed-racket/typed-racket.rkt:24:4
   /Users/mflatt/build/macro/racket/collects/racket/private/more-scheme.rkt:148:2: call-with-break-parameterization
   /Users/mflatt/build/macro/build/user/6.2.900.4/pkgs/sandbox-lib/racket/sandbox.rkt:837:5: loop
Type Checker: Summary: 3 errors encountered

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 [[7 5] [3 -2]]))
- : (Array Real)
(array #[#[2/29 5/29] #[3/29 -7/29]])
> (matrix-inverse (matrix [[1 2] [10 20]]))
matrix-inverse: contract violation
expected: matrix-invertible?
given: (array #[#[1 2] #[10 20]])
> (matrix-inverse (matrix [[1 2] [10 20]]) (λ () #f))
- : (U False (Array Real))
#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 '(1 2 3 4)))
- : Real
24
> (* 1 2 3 4)
- : Integer [more precisely: Positive-Integer]
24
> (matrix-determinant (matrix [[1 2] [10 20]]))
- : Real
0
> (matrix-determinant (col-matrix [1 2]))
square-matrix-size: contract violation
expected: square-matrix?
given: (array #[#[1] #[2]])

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1	Constants and Elementary Functions
2	Flonums
3	Special Functions
4	Number Theory
5	Arbitrary-Precision Floating-Point Numbers (Bigfloats)
6	Arrays
7	Matrices and Linear Algebra
8	Statistics Functions
9	Probability Distributions
10	Stuff That Doesn’t Belong Anywhere Else

7.1	Introduction
7.2	Types, Predicates and Accessors
7.3	Construction
7.4	Conversion
7.5	Entrywise Operations and Arithmetic
7.6	Polymorphic Operations
7.7	Basic Operations
7.8	Inner Product Space Operations
7.9	Solving Systems of Equations
7.10	Row-Based Algorithms
7.11	Orthogonal Algorithms
7.12	Operator Norms and Comparing Matrices