Linear Algebra

# Linear Algebra ## Standard Functions Linear algebra functions in Julia are largely implemented by calling functions from [LAPACK](http://www.netlib.org/lapack/). Sparse factorizations call functions from [SuiteSparse](http://faculty.cse.tamu.edu/davis/suitesparse.html). ```@docs Base.:*(::AbstractArray, ::AbstractArray) Base.:\(::AbstractArray, ::Any) Base.LinAlg.dot Base.LinAlg.vecdot Base.LinAlg.cross Base.LinAlg.factorize Base.LinAlg.Diagonal Base.LinAlg.Bidiagonal Base.LinAlg.SymTridiagonal Base.LinAlg.Tridiagonal Base.LinAlg.Symmetric Base.LinAlg.Hermitian Base.LinAlg.lu Base.LinAlg.lufact Base.LinAlg.lufact! Base.LinAlg.chol Base.LinAlg.cholfact Base.LinAlg.cholfact! Base.LinAlg.lowrankupdate Base.LinAlg.lowrankdowndate Base.LinAlg.lowrankupdate! Base.LinAlg.lowrankdowndate! Base.LinAlg.ldltfact Base.LinAlg.ldltfact! Base.LinAlg.qr Base.LinAlg.qr! Base.LinAlg.qrfact Base.LinAlg.qrfact! Base.LinAlg.QR Base.LinAlg.QRCompactWY Base.LinAlg.QRPivoted Base.LinAlg.lqfact! Base.LinAlg.lqfact Base.LinAlg.lq Base.LinAlg.bkfact Base.LinAlg.bkfact! Base.LinAlg.eig Base.LinAlg.eigvals Base.LinAlg.eigvals! Base.LinAlg.eigmax Base.LinAlg.eigmin Base.LinAlg.eigvecs Base.LinAlg.eigfact Base.LinAlg.eigfact! Base.LinAlg.hessfact Base.LinAlg.hessfact! Base.LinAlg.schurfact Base.LinAlg.schurfact! Base.LinAlg.schur Base.LinAlg.ordschur Base.LinAlg.ordschur! Base.LinAlg.svdfact Base.LinAlg.svdfact! Base.LinAlg.svd Base.LinAlg.svdvals Base.LinAlg.Givens Base.LinAlg.givens Base.LinAlg.triu Base.LinAlg.triu! Base.LinAlg.tril Base.LinAlg.tril! Base.LinAlg.diagind Base.LinAlg.diag Base.LinAlg.diagm Base.LinAlg.scale! Base.LinAlg.rank Base.LinAlg.norm Base.LinAlg.vecnorm Base.LinAlg.normalize! Base.LinAlg.normalize Base.LinAlg.cond Base.LinAlg.condskeel Base.LinAlg.trace Base.LinAlg.det Base.LinAlg.logdet Base.LinAlg.logabsdet Base.inv Base.LinAlg.pinv Base.LinAlg.nullspace Base.repmat Base.repeat Base.kron Base.SparseArrays.blkdiag Base.LinAlg.linreg Base.LinAlg.expm Base.LinAlg.logm Base.LinAlg.sqrtm Base.LinAlg.lyap Base.LinAlg.sylvester Base.LinAlg.issymmetric Base.LinAlg.isposdef Base.LinAlg.isposdef! Base.LinAlg.istril Base.LinAlg.istriu Base.LinAlg.isdiag Base.LinAlg.ishermitian Base.LinAlg.RowVector Base.LinAlg.ConjArray Base.transpose Base.transpose! Base.ctranspose Base.ctranspose! Base.LinAlg.eigs(::Any) Base.LinAlg.eigs(::Any, ::Any) Base.LinAlg.svds Base.LinAlg.peakflops ``` ## Low-level matrix operations Matrix operations involving transpositions operations like `A' \ B` are converted by the Julia parser into calls to specially named functions like [`Ac_ldiv_B`](@ref). If you want to overload these operations for your own types, then it is useful to know the names of these functions. Also, in many cases there are in-place versions of matrix operations that allow you to supply a pre-allocated output vector or matrix. This is useful when optimizing critical code in order to avoid the overhead of repeated allocations. These in-place operations are suffixed with `!` below (e.g. [`A_mul_B!`](@ref)) according to the usual Julia convention. ```@docs Base.LinAlg.A_ldiv_B! Base.A_ldiv_Bc Base.A_ldiv_Bt Base.LinAlg.A_mul_B! Base.A_mul_Bc Base.A_mul_Bt Base.A_rdiv_Bc Base.A_rdiv_Bt Base.Ac_ldiv_B Base.LinAlg.Ac_ldiv_B! Base.Ac_ldiv_Bc Base.Ac_mul_B Base.Ac_mul_Bc Base.Ac_rdiv_B Base.Ac_rdiv_Bc Base.At_ldiv_B Base.LinAlg.At_ldiv_B! Base.At_ldiv_Bt Base.At_mul_B Base.At_mul_Bt Base.At_rdiv_B Base.At_rdiv_Bt ``` ## BLAS Functions In Julia (as in much of scientific computation), dense linear-algebra operations are based on the [LAPACK library](http://www.netlib.org/lapack/), which in turn is built on top of basic linear-algebra building-blocks known as the [BLAS](http://www.netlib.org/blas/). There are highly optimized implementations of BLAS available for every computer architecture, and sometimes in high-performance linear algebra routines it is useful to call the BLAS functions directly. `Base.LinAlg.BLAS` provides wrappers for some of the BLAS functions. Those BLAS functions that overwrite one of the input arrays have names ending in `'!'`. Usually, a BLAS function has four methods defined, for `Float64`, `Float32`, `Complex128`, and `Complex64` arrays. ### [BLAS Character Arguments](@id stdlib-blas-chars) Many BLAS functions accept arguments that determine whether to transpose an argument (`trans`), which triangle of a matrix to reference (`uplo` or `ul`), whether the diagonal of a triangular matrix can be assumed to be all ones (`dA`) or which side of a matrix multiplication the input argument belongs on (`side`). The possiblities are: #### [Multplication Order](@id stdlib-blas-side) | `side` | Meaning | |:-------|:--------------------------------------------------------------------| | `'L'` | The argument goes on the *left* side of a matrix-matrix operation. | | `'R'` | The argument goes on the *right* side of a matrix-matrix operation. | #### [Triangle Referencing](@id stdlib-blas-uplo) | `uplo`/`ul` | Meaning | |:------------|:------------------------------------------------------| | `'U'` | Only the *upper* triangle of the matrix will be used. | | `'L'` | Only the *lower* triangle of the matrix will be used. | #### [Transposition Operation](@id stdlib-blas-trans) | `trans`/`tX` | Meaning | |:-------------|:--------------------------------------------------------| | `'N'` | The input matrix `X` is not transposed or conjugated. | | `'T'` | The input matrix `X` will be transposed. | | `'C'` | The input matrix `X` will be conjugated and transposed. | #### [Unit Diagonal](@id stdlib-blas-diag) | `diag`/`dX` | Meaning | |:------------|:----------------------------------------------------------| | `'N'` | The diagonal values of the matrix `X` will be read. | | `'U'` | The diagonal of the matrix `X` is assumed to be all ones. | ```@docs Base.LinAlg.BLAS.dotu Base.LinAlg.BLAS.dotc Base.LinAlg.BLAS.blascopy! Base.LinAlg.BLAS.nrm2 Base.LinAlg.BLAS.asum Base.LinAlg.axpy! Base.LinAlg.BLAS.scal! Base.LinAlg.BLAS.scal Base.LinAlg.BLAS.ger! Base.LinAlg.BLAS.syr! Base.LinAlg.BLAS.syrk! Base.LinAlg.BLAS.syrk Base.LinAlg.BLAS.her! Base.LinAlg.BLAS.herk! Base.LinAlg.BLAS.herk Base.LinAlg.BLAS.gbmv! Base.LinAlg.BLAS.gbmv Base.LinAlg.BLAS.sbmv! Base.LinAlg.BLAS.sbmv(::Any, ::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.sbmv(::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.gemm! Base.LinAlg.BLAS.gemm(::Any, ::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.gemm(::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.gemv! Base.LinAlg.BLAS.gemv(::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.gemv(::Any, ::Any, ::Any) Base.LinAlg.BLAS.symm! Base.LinAlg.BLAS.symm(::Any, ::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.symm(::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.symm(::Char, ::Char, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.symv! Base.LinAlg.BLAS.symv(::Any, ::Any, ::Any, ::Any) Base.LinAlg.BLAS.symv(::Any, ::Any, ::Any) Base.LinAlg.BLAS.trmm! Base.LinAlg.BLAS.trmm Base.LinAlg.BLAS.trsm! Base.LinAlg.BLAS.trsm Base.LinAlg.BLAS.trmv! Base.LinAlg.BLAS.trmv Base.LinAlg.BLAS.trsv! Base.LinAlg.BLAS.trsv Base.LinAlg.BLAS.set_num_threads Base.LinAlg.I ``` ## LAPACK Functions `Base.LinAlg.LAPACK` provides wrappers for some of the LAPACK functions for linear algebra. Those functions that overwrite one of the input arrays have names ending in `'!'`. Usually a function has 4 methods defined, one each for `Float64`, `Float32`, `Complex128` and `Complex64` arrays. Note that the LAPACK API provided by Julia can and will change in the future. Since this API is not user-facing, there is no commitment to support/deprecate this specific set of functions in future releases. ```@docs Base.LinAlg.LAPACK.gbtrf! Base.LinAlg.LAPACK.gbtrs! Base.LinAlg.LAPACK.gebal! Base.LinAlg.LAPACK.gebak! Base.LinAlg.LAPACK.gebrd! Base.LinAlg.LAPACK.gelqf! Base.LinAlg.LAPACK.geqlf! Base.LinAlg.LAPACK.geqrf! Base.LinAlg.LAPACK.geqp3! Base.LinAlg.LAPACK.gerqf! Base.LinAlg.LAPACK.geqrt! Base.LinAlg.LAPACK.geqrt3! Base.LinAlg.LAPACK.getrf! Base.LinAlg.LAPACK.tzrzf! Base.LinAlg.LAPACK.ormrz! Base.LinAlg.LAPACK.gels! Base.LinAlg.LAPACK.gesv! Base.LinAlg.LAPACK.getrs! Base.LinAlg.LAPACK.getri! Base.LinAlg.LAPACK.gesvx! Base.LinAlg.LAPACK.gelsd! Base.LinAlg.LAPACK.gelsy! Base.LinAlg.LAPACK.gglse! Base.LinAlg.LAPACK.geev! Base.LinAlg.LAPACK.gesdd! Base.LinAlg.LAPACK.gesvd! Base.LinAlg.LAPACK.ggsvd! Base.LinAlg.LAPACK.ggsvd3! Base.LinAlg.LAPACK.geevx! Base.LinAlg.LAPACK.ggev! Base.LinAlg.LAPACK.gtsv! Base.LinAlg.LAPACK.gttrf! Base.LinAlg.LAPACK.gttrs! Base.LinAlg.LAPACK.orglq! Base.LinAlg.LAPACK.orgqr! Base.LinAlg.LAPACK.orgql! Base.LinAlg.LAPACK.orgrq! Base.LinAlg.LAPACK.ormlq! Base.LinAlg.LAPACK.ormqr! Base.LinAlg.LAPACK.ormql! Base.LinAlg.LAPACK.ormrq! Base.LinAlg.LAPACK.gemqrt! Base.LinAlg.LAPACK.posv! Base.LinAlg.LAPACK.potrf! Base.LinAlg.LAPACK.potri! Base.LinAlg.LAPACK.potrs! Base.LinAlg.LAPACK.pstrf! Base.LinAlg.LAPACK.ptsv! Base.LinAlg.LAPACK.pttrf! Base.LinAlg.LAPACK.pttrs! Base.LinAlg.LAPACK.trtri! Base.LinAlg.LAPACK.trtrs! Base.LinAlg.LAPACK.trcon! Base.LinAlg.LAPACK.trevc! Base.LinAlg.LAPACK.trrfs! Base.LinAlg.LAPACK.stev! Base.LinAlg.LAPACK.stebz! Base.LinAlg.LAPACK.stegr! Base.LinAlg.LAPACK.stein! Base.LinAlg.LAPACK.syconv! Base.LinAlg.LAPACK.sysv! Base.LinAlg.LAPACK.sytrf! Base.LinAlg.LAPACK.sytri! Base.LinAlg.LAPACK.sytrs! Base.LinAlg.LAPACK.hesv! Base.LinAlg.LAPACK.hetrf! Base.LinAlg.LAPACK.hetri! Base.LinAlg.LAPACK.hetrs! Base.LinAlg.LAPACK.syev! Base.LinAlg.LAPACK.syevr! Base.LinAlg.LAPACK.sygvd! Base.LinAlg.LAPACK.bdsqr! Base.LinAlg.LAPACK.bdsdc! Base.LinAlg.LAPACK.gecon! Base.LinAlg.LAPACK.gehrd! Base.LinAlg.LAPACK.orghr! Base.LinAlg.LAPACK.gees! Base.LinAlg.LAPACK.gges! Base.LinAlg.LAPACK.trexc! Base.LinAlg.LAPACK.trsen! Base.LinAlg.LAPACK.tgsen! Base.LinAlg.LAPACK.trsyl! ```

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