Core mathematical protocols for matrix algebra, decompositions, and quantum state operations.
Provides a contract for implementing backends to support linear algebra operations over numeric fields (ℝ or ℂ) with a focus on immutability and pure functions.
Core mathematical protocols for matrix algebra, decompositions, and quantum state operations. Provides a contract for implementing backends to support linear algebra operations over numeric fields (ℝ or ℂ) with a focus on immutability and pure functions.
Protocol for converting between QClojure and backend-specific representations.
Enables seamless interoperability between different mathematical backends while maintaining a consistent API at the QClojure level.
Protocol for converting between QClojure and backend-specific representations. Enables seamless interoperability between different mathematical backends while maintaining a consistent API at the QClojure level.
(backend->matrix backend m)Convert a backend matrix to QClojure representation.
Parameters:
Returns: QClojure matrix (vector of row vectors or complex map)
Convert a backend matrix to QClojure representation. Parameters: - backend: Backend instance - m: Matrix in backend's native format Returns: QClojure matrix (vector of row vectors or complex map)
(backend->scalar backend s)Convert a backend scalar to QClojure representation.
Parameters:
Returns: QClojure scalar (number or Vec2 complex)
Convert a backend scalar to QClojure representation. Parameters: - backend: Backend instance - s: Scalar in backend's native format Returns: QClojure scalar (number or Vec2 complex)
(backend->vector backend v)Convert a backend vector to QClojure representation.
Parameters:
Returns: QClojure vector (sequence of numbers or complex map)
Convert a backend vector to QClojure representation. Parameters: - backend: Backend instance - v: Vector in backend's native format Returns: QClojure vector (sequence of numbers or complex map)
(matrix->backend backend m)Convert a QClojure matrix to backend-specific representation.
Parameters:
Returns: Matrix in the backend's native format
Convert a QClojure matrix to backend-specific representation. Parameters: - backend: Backend instance - m: QClojure matrix (vector of row vectors or complex map) Returns: Matrix in the backend's native format
(scalar->backend backend s)Convert a QClojure scalar to backend-specific representation.
Parameters:
Returns: Scalar in the backend's native format
Convert a QClojure scalar to backend-specific representation. Parameters: - backend: Backend instance - s: QClojure scalar (number or Vec2 complex) Returns: Scalar in the backend's native format
(vector->backend backend v)Convert a QClojure vector to backend-specific representation.
Parameters:
Returns: Vector in the backend's native format
Convert a QClojure vector to backend-specific representation. Parameters: - backend: Backend instance - v: QClojure vector (sequence of numbers or complex map) Returns: Vector in the backend's native format
Backend contract for core linear algebra operations over a numeric field.
Rationale:
Conventions used in docstrings below:
Implementations may return concrete Clojure data (vectors of vectors, maps with :real/:imag parts, records, or opaque handles). Callers should treat returned values as opaque and use protocol functions for further manipulation.
All functions are expected to be pure with respect to their arguments (i.e. no in-place mutation visible to the caller) unless the backend documents an optimization that preserves semantic immutability (copy-on-write, etc.).
Backend contract for core linear algebra operations over a numeric field. Rationale: - Backends decide representation (pure real, split-complex/SoA, AoS, native lib). - Public semantics expressed in mathematical terms, not data shape. - Mixed real/complex promotion is an implementation concern (outside protocol). Conventions used in docstrings below: - Matrix : rectangular 2-D array (m x n) over a numeric field (ℝ or ℂ). - Vector : 1-D array length n over same field. - Scalar : element of the field. - Aᴴ : conjugate transpose (Hermitian adjoint). For real matrices Aᴴ = Aᵀ. - ⊙ : Hadamard (element-wise) product. - ⊗ : Kronecker (tensor) product. Implementations may return concrete Clojure data (vectors of vectors, maps with :real/:imag parts, records, or opaque handles). Callers should treat returned values as opaque and use protocol functions for further manipulation. All functions are expected to be pure with respect to their arguments (i.e. no in-place mutation visible to the caller) unless the backend documents an optimization that preserves semantic immutability (copy-on-write, etc.).
(positive-semidefinite? backend A)Test if a matrix is positive semidefinite.
Parameters:
Returns: Boolean indicating whether all eigenvalues of A are ≥ -tolerance
Note: Matrix should be Hermitian for meaningful results
Test if a matrix is positive semidefinite. Parameters: - backend: Backend instance - A: Square matrix (real or complex), should be Hermitian Returns: Boolean indicating whether all eigenvalues of A are ≥ -tolerance Note: Matrix should be Hermitian for meaningful results
(subtract backend A B)Perform matrix subtraction A - B.
Parameters:
Returns: New matrix representing A - B
Perform matrix subtraction A - B. Parameters: - backend: Backend instance - A: First matrix (real or complex) - B: Second matrix (real or complex) with same dimensions as A Returns: New matrix representing A - B
(add backend A B)Perform matrix addition A + B.
Parameters:
Returns: New matrix representing A + B
Perform matrix addition A + B. Parameters: - backend: Backend instance - A: First matrix (real or complex) - B: Second matrix (real or complex) with same dimensions as A Returns: New matrix representing A + B
(negate backend A)Compute the additive inverse -A.
Parameters:
Returns: New matrix representing -A
Note: Equivalent to (scale backend A -1) but may be optimized
Compute the additive inverse -A. Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: New matrix representing -A Note: Equivalent to (scale backend A -1) but may be optimized
(kronecker-product backend A B)Compute the Kronecker (tensor) product A ⊗ B.
Parameters:
Returns: New matrix (mp×nq) representing the Kronecker product A ⊗ B
Note: Essential for quantum computing multi-qubit operations
Compute the Kronecker (tensor) product A ⊗ B. Parameters: - backend: Backend instance - A: First matrix (m×n, real or complex) - B: Second matrix (p×q, real or complex) Returns: New matrix (mp×nq) representing the Kronecker product A ⊗ B Note: Essential for quantum computing multi-qubit operations
(hermitian? backend A)(hermitian? backend A eps)Test if a matrix is Hermitian (A ≈ Aᴴ).
Parameters:
Returns: Boolean indicating whether A is Hermitian (or real symmetric)
Test if a matrix is Hermitian (A ≈ Aᴴ). Parameters: - backend: Backend instance - A: Square matrix (real or complex) - eps: Optional tolerance for approximate equality (uses backend default if not provided) Returns: Boolean indicating whether A is Hermitian (or real symmetric)
(hadamard-product backend A B)Compute the element-wise (Hadamard) product A ⊙ B.
Parameters:
Returns: New matrix with element-wise multiplication of A and B
Compute the element-wise (Hadamard) product A ⊙ B. Parameters: - backend: Backend instance - A: First matrix (real or complex) - B: Second matrix (real or complex) with same dimensions as A Returns: New matrix with element-wise multiplication of A and B
(norm2 backend x)Compute the Euclidean (L2) norm ||x||₂.
Parameters:
Returns: Non-negative real number representing ||x||₂ = sqrt(⟨x|x⟩)
Compute the Euclidean (L2) norm ||x||₂. Parameters: - backend: Backend instance - x: Vector (real or complex) Returns: Non-negative real number representing ||x||₂ = sqrt(⟨x|x⟩)
(scale backend A alpha)Perform scalar multiplication alpha · A.
Parameters:
Returns: New matrix representing alpha · A
Perform scalar multiplication alpha · A. Parameters: - backend: Backend instance - A: Matrix (real or complex) - alpha: Scalar value (real or complex number) Returns: New matrix representing alpha · A
(inner-product backend x y)Compute the vector inner product ⟨x|y⟩.
Parameters:
Returns: Scalar representing the inner product
Note: Conjugates the first argument if complex (follows Dirac bra-ket notation)
Compute the vector inner product ⟨x|y⟩. Parameters: - backend: Backend instance - x: First vector (real or complex) - y: Second vector (real or complex) with same length as x Returns: Scalar representing the inner product Note: Conjugates the first argument if complex (follows Dirac bra-ket notation)
(inverse backend A)Compute the matrix inverse A⁻¹.
Parameters:
Returns: Inverse matrix A⁻¹, or nil if A is singular
Compute the matrix inverse A⁻¹. Parameters: - backend: Backend instance - A: Square non-singular matrix (real or complex) Returns: Inverse matrix A⁻¹, or nil if A is singular
(transpose backend A)Compute the transpose Aᵀ.
Parameters:
Returns: New matrix representing the transpose of A (swaps rows/cols without conjugation)
Compute the transpose Aᵀ. Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: New matrix representing the transpose of A (swaps rows/cols without conjugation)
(matrix-multiply backend A B)Perform matrix multiplication A × B.
Parameters:
Returns: New matrix (m×n) representing the product A × B
Perform matrix multiplication A × B. Parameters: - backend: Backend instance - A: Left matrix (m×k, real or complex) - B: Right matrix (k×n, real or complex) with compatible dimensions Returns: New matrix (m×n) representing the product A × B
(trace backend A)Compute the trace Tr(A) = Σᵢ aᵢᵢ.
Parameters:
Returns: Scalar (real or complex) representing the sum of diagonal elements
Compute the trace Tr(A) = Σᵢ aᵢᵢ. Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Scalar (real or complex) representing the sum of diagonal elements
(outer-product backend x y)Compute the outer product x ⊗ y†.
Parameters:
Returns: New matrix (m×n) representing x ⊗ y†
Note: For complex vectors, conjugates y but not x (follows Dirac bra-ket notation)
Compute the outer product x ⊗ y†. Parameters: - backend: Backend instance - x: First vector (length m, real or complex) - y: Second vector (length n, real or complex) Returns: New matrix (m×n) representing x ⊗ y† Note: For complex vectors, conjugates y but not x (follows Dirac bra-ket notation)
(conjugate-transpose backend A)Compute the conjugate transpose Aᴴ (Hermitian adjoint).
Parameters:
Returns: New matrix representing the conjugate transpose of A
Note: For real matrices, this is equivalent to transpose
Compute the conjugate transpose Aᴴ (Hermitian adjoint). Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: New matrix representing the conjugate transpose of A Note: For real matrices, this is equivalent to transpose
(matrix-vector-product backend A x)Perform matrix–vector multiplication A × x.
Parameters:
Returns: New vector (length m) representing the product A × x
Perform matrix–vector multiplication A × x. Parameters: - backend: Backend instance - A: Matrix (m×n, real or complex) - x: Vector (length n, real or complex) with compatible dimensions Returns: New vector (length m) representing the product A × x
(unitary? backend U)(unitary? backend U eps)Test if a matrix is unitary (Uᴴ U ≈ I).
Parameters:
Returns: Boolean indicating whether U is unitary (or orthogonal for real matrices)
Test if a matrix is unitary (Uᴴ U ≈ I). Parameters: - backend: Backend instance - U: Square matrix (real or complex) - eps: Optional tolerance for approximate equality (uses backend default if not provided) Returns: Boolean indicating whether U is unitary (or orthogonal for real matrices)
(shape backend A)Return the dimensions of a matrix.
Parameters:
Returns: Vector [rows cols] indicating matrix dimensions
Return the dimensions of a matrix. Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: Vector [rows cols] indicating matrix dimensions
(solve-linear-system backend A b)Solve the linear system A x = b.
Parameters:
Returns: Solution vector x or matrix, or nil/throws if A is singular or ill-conditioned
Solve the linear system A x = b. Parameters: - backend: Backend instance - A: Square matrix (n×n, real or complex), must be non-singular - b: Right-hand side vector (length n) or matrix Returns: Solution vector x or matrix, or nil/throws if A is singular or ill-conditioned
Protocol for matrix analysis and conditioning metrics.
Provides scalar-valued analyses that measure various properties of matrices, particularly useful for numerical stability assessments.
Protocol for matrix analysis and conditioning metrics. Provides scalar-valued analyses that measure various properties of matrices, particularly useful for numerical stability assessments.
(condition-number backend A)Compute the 2-norm condition number κ₂(A).
Parameters:
Returns: Positive real number representing κ₂(A) = σ_max / σ_min
Note: Higher values indicate numerical instability in linear system solutions
Compute the 2-norm condition number κ₂(A). Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Positive real number representing κ₂(A) = σ_max / σ_min Note: Higher values indicate numerical instability in linear system solutions
(spectral-norm backend A)Compute the spectral norm ||A||₂ (largest singular value).
Parameters:
Returns: Non-negative real number representing the largest singular value σ₀
Compute the spectral norm ||A||₂ (largest singular value). Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: Non-negative real number representing the largest singular value σ₀
Protocol for matrix decomposition operations.
Provides standard numerical linear algebra decompositions that are fundamental for many computational applications, especially in quantum computing.
Protocol for matrix decomposition operations. Provides standard numerical linear algebra decompositions that are fundamental for many computational applications, especially in quantum computing.
(cholesky-decomposition backend A)Compute the Cholesky decomposition A = L Lᴴ.
Parameters:
Returns: Map containing:
Note: May throw or return nil if A is not positive semidefinite
Compute the Cholesky decomposition A = L Lᴴ. Parameters: - backend: Backend instance - A: Positive semidefinite matrix (real or complex) Returns: Map containing: - :L - Lower triangular matrix such that A = L Lᴴ Note: May throw or return nil if A is not positive semidefinite
(eigen-general backend A)Compute eigenvalues for a general (possibly non-Hermitian) matrix.
Parameters:
Returns: Map containing eigenvalue information (backend-specific format)
Note: May throw if the backend doesn't support general eigenvalue computation
Compute eigenvalues for a general (possibly non-Hermitian) matrix. Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Map containing eigenvalue information (backend-specific format) Note: May throw if the backend doesn't support general eigenvalue computation
(eigen-hermitian backend A)Compute the eigendecomposition of a Hermitian matrix.
Parameters:
Returns: Map containing:
Compute the eigendecomposition of a Hermitian matrix. Parameters: - backend: Backend instance - A: Square Hermitian matrix (real symmetric or complex Hermitian) Returns: Map containing: - :eigenvalues - Vector of eigenvalues in ascending order [λ₀≤...≤λₙ₋₁] - :eigenvectors - Vector of corresponding eigenvector columns [v₀ ...]
(lu-decomposition backend A)Compute the LU decomposition A = P L U.
Parameters:
Returns: Map containing:
Compute the LU decomposition A = P L U. Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Map containing: - :P - Row permutation matrix - :L - Lower triangular matrix - :U - Upper triangular matrix
(qr-decomposition backend A)Compute the QR decomposition A = Q R.
Parameters:
Returns: Map containing:
Note: Backend may return additional components depending on implementation
Compute the QR decomposition A = Q R. Parameters: - backend: Backend instance - A: Matrix (real or complex) Returns: Map containing: - :Q - Orthogonal/unitary matrix - :R - Upper triangular matrix Note: Backend may return additional components depending on implementation
(svd backend A)Compute the Singular Value Decomposition A = U Σ Vᴴ.
Parameters:
Returns: Map containing:
Compute the Singular Value Decomposition A = U Σ Vᴴ. Parameters: - backend: Backend instance - A: Matrix (real or complex, may be rectangular) Returns: Map containing: - :U - Left singular vectors matrix - :S - Vector of singular values in descending order - :V† - Right singular vectors conjugate transpose matrix
Protocol for analytic functions of matrices.
Provides matrix-valued functions using functional calculus. Implementations may use various approximation methods such as Padé approximants, scaling & squaring, or Schur decomposition.
Protocol for analytic functions of matrices. Provides matrix-valued functions using functional calculus. Implementations may use various approximation methods such as Padé approximants, scaling & squaring, or Schur decomposition.
(matrix-exp backend A)Compute the matrix exponential exp(A).
Parameters:
Returns: Matrix representing exp(A)
Note: Important for quantum time evolution operators and solving differential equations
Compute the matrix exponential exp(A). Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Matrix representing exp(A) Note: Important for quantum time evolution operators and solving differential equations
(matrix-log backend A)Compute the principal matrix logarithm log(A).
Parameters:
Returns: Matrix representing the principal branch of log(A)
Note: Domain restrictions apply for matrices with negative or zero eigenvalues
Compute the principal matrix logarithm log(A). Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Matrix representing the principal branch of log(A) Note: Domain restrictions apply for matrices with negative or zero eigenvalues
(matrix-sqrt backend A)Compute the principal matrix square root √A.
Parameters:
Returns: Matrix representing the principal square root of A
Note: Best defined for positive semidefinite matrices
Compute the principal matrix square root √A. Parameters: - backend: Backend instance - A: Square matrix (real or complex) Returns: Matrix representing the principal square root of A Note: Best defined for positive semidefinite matrices
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