1"use strict";(self.webpackChunkrapier=self.webpackChunkrapier||[]).push([[764],{8064:(s,e,a)=>{a.r(e),a.d(e,{assets:()=>h,contentTitle:()=>m,default:()=>x,frontMatter:()=>c,metadata:()=>n,toc:()=>d});const n=JSON.parse('{"id":"user_guides/templates/wgebra/matrix_decompositions","title":"Matrix decompositions","description":"Automatic WGSL docs generation publishable on this website is currently waiting on wgpu#6364.","source":"@site/docs/user_guides/templates/wgebra/matrix_decompositions.mdx","sourceDirName":"user_guides/templates/wgebra","slug":"/user_guides/templates/wgebra/matrix_decompositions","permalink":"/docs/user_guides/templates/wgebra/matrix_decompositions","draft":false,"unlisted":false,"tags":[],"version":"current","frontMatter":{"id":"matrix_decompositions","title":"Matrix decompositions","sidebar_label":"Matrix decompositions"}}');var t=a(4848),i=a(8453),l=a(1470),r=a(9365);const c={id:"matrix_decompositions",title:"Matrix decompositions",sidebar_label:"Matrix decompositions"},m=void 0,h={},d=[{value:"Cholesky",id:"cholesky",level:2},{value:"QR",id:"qr",level:2},{value:"LU with partial pivoting",id:"lu-with-partial-pivoting",level:2},{value:"Eigendecomposition (symmetric matrix)",id:"eigendecomposition-symmetric-matrix",level:2},{value:"Singular Value Decomposition",id:"singular-value-decomposition",level:2}];function o(s){const e={a:"a",admonition:"admonition",annotation:"annotation",code:"code",em:"em",h2:"h2",img:"img",math:"math",mi:"mi",mn:"mn",mo:"mo",mrow:"mrow",msup:"msup",mtext:"mtext",p:"p",pre:"pre",semantics:"semantics",span:"span",strong:"strong",table:"table",tbody:"tbody",td:"td",th:"th",thead:"thead",tr:"tr",...(0,i.R)(),...s.components};return(0,t.jsxs)(t.Fragment,{children:[(0,t.jsx)(e.admonition,{type:"warning",children:(0,t.jsxs)(e.p,{children:["Automatic WGSL docs generation publishable on this website is currently waiting on ",(0,t.jsx)(e.a,{href:"https://github.com/gfx-rs/wgpu/pull/6364",children:"wgpu#6364"}),".\nIn the meantime, refer to the WGSL sources in the ",(0,t.jsx)(e.a,{href:"https://github.com/dimforge/wgmath/tree/main/crates/wgebra",children:"wgebra repository"}),"."]})}),"\n",(0,t.jsx)(e.p,{children:"Matrix decomposition is a family of techniques that aim to represent a matrix as\nthe product of several matrices. Those factors can either allow more efficient\noperations like inversion or linear system resolution, and might provide some\ninsight regarding intrinsic properties of some data to be analysed (e.g. by\nobserving singular values, eigenvectors, etc.)"}),"\n",(0,t.jsx)(e.admonition,{type:"info",children:(0,t.jsxs)(e.p,{children:[(0,t.jsx)(e.strong,{children:"WGebra"})," implements common matrix decompositions for low-dimensional matrices (currently ",(0,t.jsx)(e.code,{children:"mat2x2<f32>"}),", ",(0,t.jsx)(e.code,{children:"mat3x3<f32>"}),"\nand ",(0,t.jsx)(e.code,{children:"mat4x4<f32>"}),"). There is no implementation of these decomposition for large matrices yet."]})}),"\n",(0,t.jsxs)(e.table,{children:[(0,t.jsx)(e.thead,{children:(0,t.jsxs)(e.tr,{children:[(0,t.jsx)(e.th,{children:"Decomposition"}),(0,t.jsx)(e.th,{children:"Factors"}),(0,t.jsx)(e.th,{children:"Derivable shaders"})]})}),(0,t.jsxs)(e.tbody,{children:[(0,t.jsxs)(e.tr,{children:[(0,t.jsx)(e.td,{children:"QR"}),(0,t.jsx)(e.td,{children:(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"Q"}),(0,t.jsx)(e.mtext,{children:"\xa0"}),(0,t.jsx)(e.mi,{children:"R"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q ~ R"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"Q"}),(0,t.jsx)(e.span,{className:"mspace nobreak",children:"\xa0"}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.00773em"},children:"R"})]})})]})}),(0,t.jsxs)(e.td,{children:[(0,t.jsx)(e.code,{children:"WgQR2"}),", ",(0,t.jsx)(e.code,{children:"WgQR3"}),", ",(0,t.jsx)(e.code,{children:"WgQR4"})]})]}),(0,t.jsxs)(e.tr,{children:[(0,t.jsx)(e.td,{children:"LU with partial pivoting"}),(0,t.jsx)(e.td,{children:(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsxs)(e.msup,{children:[(0,t.jsx)(e.mi,{children:"P"}),(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mo,{children:"\u2212"}),(0,t.jsx)(e.mn,{children:"1"})]})]}),(0,t.jsx)(e.mtext,{children:"\xa0"}),(0,t.jsx)(e.mi,{children:"L"}),(0,t.jsx)(e.mtext,{children:"\xa0"}),(0,t.jsx)(e.mi,{children:"U"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"P^{-1} ~ L ~ U"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.8141em"}}),(0,t.jsxs)(e.span,{className:"mord",children:[(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"}),(0,t.jsx)(e.span,{className:"msupsub",children:(0,t.jsx)(e.span,{className:"vlist-t",children:(0,t.jsx)(e.span,{className:"vlist-r",children:(0,t.jsx)(e.span,{className:"vlist",style:{height:"0.8141em"}
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1,children:(0,t.jsxs)(e.span,{style:{top:"-3.063em",marginRight:"0.05em"},children:[(0,t.jsx)(e.span,{className:"pstrut",style:{height:"2.7em"}}),(0,t.jsx)(e.span,{className:"sizing reset-size6 size3 mtight",children:(0,t.jsx)(e.span,{className:"mbin mtight",children:"\u2217"})})]})})})})})]})]})})]})," designates the conjugate-transpose of ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"L"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"L"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"L"})]})})]}),". If the input matrix is\nnot SDP, such a decomposition does not exist and the matrix method\n",(0,t.jsx)(e.code,{children:".cholesky(...)"})," returns ",(0,t.jsx)(e.code,{children:"None"}),". Note that the input matrix is interpreted as a\nhermitian matrix. Only its lower-triangular part (including the diagonal) is\nread by the Cholesky decomposition (its strictly upper-triangular is never\naccessed in memory). See ",(0,t.jsx)(e.a,{href:"https://en.wikipedia.org/wiki/Cholesky_decomposition",children:"the wikipedia\narticle"})," for further\ndetails about the Cholesky decomposition."]}),"\n",(0,t.jsx)(e.p,{children:"Typical uses of the Cholesky decomposition include the inversion of SDP\nmatrices and resolution of SDP linear systems."}),"\n",(0,t.jsx)("div",{style:{textAlign:"center"},children:(0,t.jsx)(e.p,{children:(0,t.jsx)(e.img,{src:"https://nalgebra.rs/assets/images/cholesky-75dc99e2e735888f6626
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general ",(0,t.jsx)(e.code,{children:"m \xd7 n"})," matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"M"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is composed of an ",(0,t.jsx)(e.code,{children:"m \xd7 min(n, m)"})," unitary matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"Q"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]}),", and a ",(0,t.jsx)(e.code,{children:"min(n, m) \xd7 m"})," upper triangular matrix\n(or upper trapezoidal if ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"m"}),(0,t.jsx)(e.mo,{children:"<"}),(0,t.jsx)(e.mi,{children:"n"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"m < n"})]})})}),(0,t.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.5782em",verticalAlign:"-0.0391em"}}),(0,t.jsx)(e.span,{className:"mord 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mathnormal",style:{marginRight:"0.00773em"},children:"R"})]})})]})," such that ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"M"}),(0,t.jsx)(e.mo,{children:"="}),(0,t.jsx)(e.mi,{children:"Q"}),(0,t.jsx)(e.mi,{children:"R"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M = QR"})]})})}),(0,t.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord 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This can be used to\ncompute the inverse or a matrix or for solving linear systems. See also ",(0,t.jsx)(e.a,{href:"https://en.wikipedia.org/wiki/QR_decomposition",children:"the\nwikipedia article"})," for further\ndetails."]}),"\n",(0,t.jsx)("div",{style:{textAlign:"center"},children:(0,t.jsx)(e.p,{children:(0,t.jsx)(e.img,{src:"https://nalgebra.rs/assets/images/QR-c637c07671e0b3b667b54e70ea195a67.svg",alt:"QR decomposition of a 3x3 matrix."})})}),"\n",(0,t.jsxs)(l.A,{groupId:"wgebra",defaultValue:"rust",values:[{label:"main.rs",value:"rust"},{label:"your_shader.wgsl",value:"wgsl"}],children:[(0,t.jsx)(r.A,{value:"rust",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:'#[derive(Shader)]\n#[shader(derive(WgQR3), src = "your_shader.wgsl")]\nstruct YourShader;\n\n// Automatically generates a test to check with `cargo test` if `your_shader.wgsl` compiles.\nwgcore::test_shader_compilation!(YourShader);\n'})})}),(0,t.jsx)(r.A,{value:"wgsl",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:"#import wgebra::qr3 as QR\n\n// Example of function running the decomposition on a 3x3 matrix.\nfn my_function(m: mat3x3<f32>) {\n let decomposition = QR::qr(m);\n}\n"})})})]}),"\n",(0,t.jsx)(e.h2,{id:"lu-with-partial-pivoting",children:"LU with partial pivoting"}),"\n",(0,t.jsxs)(e.p,{children:["The LU decomposition of a general ",(0,t.jsx)(e.code,{children:"m \xd7 n"})," matrix is composed of a ",(0,t.jsx)(e.code,{children:"m \xd7 min(n, m)"})," lower triangular matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"L"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"L"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"L"})]})})]})," with a diagonal filled with 1, and a ",(0,t.jsx)(e.code,{children:"min(n, m) \xd7 m"})," upper triangular matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"U"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"U"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"U"})]})})]})," such that ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"M"}),(0,t.jsx)(e.mo,{children:"="}),(0,t.jsx)(e.mi,{children:"L"}),(0,t.jsx)(e.mi,{children:"U"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M = LU"})]})})}),(0,t.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"}),(0,t.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}}),(0,t.jsx)(e.span,{className:"mrel",children:"="}),(0,t.jsx)(e.span,{className:"mspace",style:{marginRight:"0.2778em"}})]}),(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"LU"})]})]})]}),". This decomposition is\ntypically used for solving linear systems, compute determinants, matrix\ninverse, and matrix rank."]}),"\n",(0,t.jsxs)(e.p,{children:["WGebra implements ",(0,t.jsx)(e.code,{children:"LU"})," decomposition with partial (row) pivoting which computes additionally\nonly one permutation matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"P"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"P"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.13889em"},children:"P"})]})})]})," such that ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"P"}),(0,t.jsx)(e.mi,{children:"M"}),(0,t.jsx)(e.mo,{children:"="}),(0,t.jsx)(e.mi,{children:"L"}),(0,t.jsx)(e.mi,{children:"U"})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"PM = LU"})]})})}),(0,t.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord 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matrix."})})}),"\n",(0,t.jsxs)(l.A,{groupId:"wgebra",defaultValue:"rust",values:[{label:"main.rs",value:"rust"},{label:"your_shader.wgsl",value:"wgsl"}],children:[(0,t.jsx)(r.A,{value:"rust",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:'#[derive(Shader)]\n#[shader(derive(WgLU3), src = "your_shader.wgsl")]\nstruct YourShader;\n\n// Automatically generates a test to check with `cargo test` if `your_shader.wgsl` compiles.\nwgcore::test_shader_compilation!(YourShader);\n'})})}),(0,t.jsx)(r.A,{value:"wgsl",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:"#import wgebra::lu3 as LU\n\n// Example of function running the decomposition on a 3x3 matrix.\nfn my_function(m: mat3x3<f32>) {\n let decomposition = LU::lu(m);\n}\n"})})})]}),"\n",(0,t.jsx)(e.h2,{id:"eigendecomposition-symmetric-matrix",children:"Eigendecomposition (symmetric matrix)"}),"\n",(0,t.jsxs)(e.p,{children:["The eigendecomposition of a square symmetric matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"M"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," is composed of an\nunitary matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"Q"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," and a real diagonal matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{mathvariant:"normal",children:"\u039b"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\Lambda"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord",children:"\u039b"})]})})]})," such that ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsxs)(e.mrow,{children:[(0,t.jsx)(e.mi,{children:"M"}),(0,t.jsx)(e.mo,{children:"="}),(0,t.jsx)(e.mi,{children:"Q"}),(0,t.jsx)(e.mi,{mathvariant:"normal",children:"\u039b"}),(0,t.jsxs)(e.msup,{children:[(0,t.jsx)(e.mi,{children:"Q"}),(0,t.jsx)(e.mo,{children:"\u2217"})]})]}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M = Q\\Lambda\nQ^*"})]})})}),(0,t.jsxs)(e.span,{className:"katex-html","aria-hidden":"true",children:[(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord 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The columns of ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"Q"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"Q"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.8778em",verticalAlign:"-0.1944em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",children:"Q"})]})})]})," are called the ",(0,t.jsx)(e.em,{children:"eigenvectors"})," of ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"M"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"M"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.10903em"},children:"M"})]})})]})," and the diagonal\nelements of 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structure."]}),"\n",(0,t.jsx)("div",{style:{textAlign:"center"},children:(0,t.jsx)(e.p,{children:(0,t.jsx)(e.img,{src:"https://nalgebra.rs/assets/images/symmetric_eigen-63031cc3f1a7d8e57497b26c16041f0b.svg",alt:"Eigendecomposition of a 3x3 symmetric matrix."})})}),"\n",(0,t.jsxs)(l.A,{groupId:"wgebra",defaultValue:"rust",values:[{label:"main.rs",value:"rust"},{label:"your_shader.wgsl",value:"wgsl"}],children:[(0,t.jsx)(r.A,{value:"rust",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:'#[derive(Shader)]\n#[shader(derive(WgSymmetricEigen3), src = "your_shader.wgsl")]\nstruct YourShader;\n\n// Automatically generates a test to check with `cargo test` if `your_shader.wgsl` compiles.\nwgcore::test_shader_compilation!(YourShader);\n'})})}),(0,t.jsx)(r.A,{value:"wgsl",children:(0,t.jsx)(e.pre,{children:(0,t.jsx)(e.code,{className:"language-rust",children:"#import wgebra::eig3 as Eig\n\n// Example of function running the decomposition on a 3x3 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",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{children:"V"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"V"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord mathnormal",style:{marginRight:"0.22222em"},children:"V"})]})})]})," and a diagonal matrix ",(0,t.jsxs)(e.span,{className:"katex",children:[(0,t.jsx)(e.span,{className:"katex-mathml",children:(0,t.jsx)(e.math,{xmlns:"http://www.w3.org/1998/Math/MathML",children:(0,t.jsxs)(e.semantics,{children:[(0,t.jsx)(e.mrow,{children:(0,t.jsx)(e.mi,{mathvariant:"normal",children:"\u03a3"})}),(0,t.jsx)(e.annotation,{encoding:"application/x-tex",children:"\\Sigma"})]})})}),(0,t.jsx)(e.span,{className:"katex-html","aria-hidden":"true",children:(0,t.jsxs)(e.span,{className:"base",children:[(0,t.jsx)(e.span,{className:"strut",style:{height:"0.6833em"}}),(0,t.jsx)(e.span,{className:"mord",children:"\u03a3"})]})})]})," with positive\n(or zero) components. Typical uses of the SVD are the pseudo-inverse, rank\ncomputation, and the resolution of least-square problems."]}),"\n",(0,t.jsxs)(e.p,{children:["The singular values, left singular vectors, and right singular vectors are\nrespectively stored on the public fields ",(0,t.jsx)(e.code,{children:"singular_values"}),", ",(0,t.jsx)(e.code,{children:"u"})," and ",(0,t.jsx)(e.code,{children:"v_t"}),". 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