Add usolveAll and lsolveAll methods (#1916)

* refactor solveValidation

* refactor usolve

* usolve algorithm implemented (for square mat.)

* added lsolve, consistent return type, fixed tests

* fixed lusolve and its tests, fixed linting issues

* added tests for usolve&lsolve, try-catch in lusolve

* put changes into separate files (u-/lsolveAll), revert changes to u-, l- and lusolve

* made *solveAll return [] for non-solvable, implemented sparse algorithms

* improved documentation for *solve(All)

Co-authored-by: Jos de Jong <wjosdejong@gmail.com>

src/expression/embeddedDocs/embeddedDocs.js

@@ -167,11 +167,13 @@ import { addDocs } from './function/arithmetic/add'
 import { absDocs } from './function/arithmetic/abs'
 import { qrDocs } from './function/algebra/qr'
 import { usolveDocs } from './function/algebra/usolve'
+import { usolveAllDocs } from './function/algebra/usolveAll'
 import { sluDocs } from './function/algebra/slu'
 import { rationalizeDocs } from './function/algebra/rationalize'
 import { simplifyDocs } from './function/algebra/simplify'
 import { lupDocs } from './function/algebra/lup'
 import { lsolveDocs } from './function/algebra/lsolve'
+import { lsolveAllDocs } from './function/algebra/lsolveAll'
 import { derivativeDocs } from './function/algebra/derivative'
 import { versionDocs } from './constants/version'
 import { trueDocs } from './constants/true'
@@ -310,12 +312,14 @@ export const embeddedDocs = {
   // functions - algebra
   derivative: derivativeDocs,
   lsolve: lsolveDocs,
+  lsolveAll: lsolveAllDocs,
   lup: lupDocs,
   lusolve: lusolveDocs,
   simplify: simplifyDocs,
   rationalize: rationalizeDocs,
   slu: sluDocs,
   usolve: usolveDocs,
+  usolveAll: usolveAllDocs,
   qr: qrDocs,
 
   // functions - arithmetic

src/expression/embeddedDocs/function/algebra/lsolve.js

@@ -5,13 +5,13 @@ export const lsolveDocs = {
     'x=lsolve(L, b)'
   ],
   description:
-  'Solves the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector.',
+  'Finds one solution of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector.',
   examples: [
     'a = [-2, 3; 2, 1]',
     'b = [11, 9]',
     'x = lsolve(a, b)'
   ],
   seealso: [
-    'lup', 'lusolve', 'usolve', 'matrix', 'sparse'
+    'lsolveAll', 'lup', 'lusolve', 'usolve', 'matrix', 'sparse'
   ]
 }

src/expression/embeddedDocs/function/algebra/lsolveAll.js

@@ -0,0 +1,17 @@
+export const lsolveAllDocs = {
+  name: 'lsolveAll',
+  category: 'Algebra',
+  syntax: [
+    'x=lsolveAll(L, b)'
+  ],
+  description:
+  'Finds all solutions of the linear system L * x = b where L is an [n x n] lower triangular matrix and b is a [n] column vector.',
+  examples: [
+    'a = [-2, 3; 2, 1]',
+    'b = [11, 9]',
+    'x = lsolve(a, b)'
+  ],
+  seealso: [
+    'lsolve', 'lup', 'lusolve', 'usolve', 'matrix', 'sparse'
+  ]
+}

src/expression/embeddedDocs/function/algebra/usolve.js

@@ -5,11 +5,11 @@ export const usolveDocs = {
     'x=usolve(U, b)'
   ],
   description:
-  'Solves the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector.',
+  'Finds one solution of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector.',
   examples: [
     'x=usolve(sparse([1, 1, 1, 1; 0, 1, 1, 1; 0, 0, 1, 1; 0, 0, 0, 1]), [1; 2; 3; 4])'
   ],
   seealso: [
-    'lup', 'lusolve', 'lsolve', 'matrix', 'sparse'
+    'usolveAll', 'lup', 'lusolve', 'lsolve', 'matrix', 'sparse'
   ]
 }

src/expression/embeddedDocs/function/algebra/usolveAll.js

@@ -0,0 +1,15 @@
+export const usolveAllDocs = {
+  name: 'usolveAll',
+  category: 'Algebra',
+  syntax: [
+    'x=usolve(U, b)'
+  ],
+  description:
+  'Finds all solutions of the linear system U * x = b where U is an [n x n] upper triangular matrix and b is a [n] column vector.',
+  examples: [
+    'x=usolve(sparse([1, 1, 1, 1; 0, 1, 1, 1; 0, 0, 1, 1; 0, 0, 0, 1]), [1; 2; 3; 4])'
+  ],
+  seealso: [
+    'usolve', 'lup', 'lusolve', 'lsolve', 'matrix', 'sparse'
+  ]
+}

src/factoriesAny.js

@@ -103,6 +103,8 @@ export { createDotPow } from './function/arithmetic/dotPow'
 export { createDotDivide } from './function/arithmetic/dotDivide'
 export { createLsolve } from './function/algebra/solver/lsolve'
 export { createUsolve } from './function/algebra/solver/usolve'
+export { createLsolveAll } from './function/algebra/solver/lsolveAll'
+export { createUsolveAll } from './function/algebra/solver/usolveAll'
 export { createLeftShift } from './function/bitwise/leftShift'
 export { createRightArithShift } from './function/bitwise/rightArithShift'
 export { createRightLogShift } from './function/bitwise/rightLogShift'

src/function/algebra/solver/lsolve.js

@@ -16,7 +16,7 @@ export const createLsolve = /* #__PURE__ */ factory(name, dependencies, ({ typed
   const solveValidation = createSolveValidation({ DenseMatrix })
 
   /**
-   * Solves the linear equation system by forwards substitution. Matrix must be a lower triangular matrix.
+   * Finds one solution of a linear equation system by forwards substitution. Matrix must be a lower triangular matrix. Throws an error if there's no solution.
    *
    * `L * x = b`
    *
@@ -32,7 +32,7 @@ export const createLsolve = /* #__PURE__ */ factory(name, dependencies, ({ typed
    *
    * See also:
    *
-   *    lup, slu, usolve, lusolve
+   *    lsolveAll, lup, slu, usolve, lusolve
    *
    * @param {Matrix, Array} L       A N x N matrix or array (L)
    * @param {Matrix, Array} b       A column vector with the b values
@@ -42,67 +42,61 @@ export const createLsolve = /* #__PURE__ */ factory(name, dependencies, ({ typed
   return typed(name, {
 
     'SparseMatrix, Array | Matrix': function (m, b) {
-      // process matrix
       return _sparseForwardSubstitution(m, b)
     },
 
     'DenseMatrix, Array | Matrix': function (m, b) {
-      // process matrix
       return _denseForwardSubstitution(m, b)
     },
 
     'Array, Array | Matrix': function (a, b) {
-      // create dense matrix from array
       const m = matrix(a)
-      // use matrix implementation
       const r = _denseForwardSubstitution(m, b)
-      // result
       return r.valueOf()
     }
   })
 
   function _denseForwardSubstitution (m, b) {
     // validate matrix and vector, return copy of column vector b
     b = solveValidation(m, b, true)
-    // column vector data
     const bdata = b._data
-    // rows & columns
+
     const rows = m._size[0]
     const columns = m._size[1]
+
     // result
     const x = []
-    // data
-    const data = m._data
-    // forward solve m * x = b, loop columns
+
+    const mdata = m._data
+
+    // loop columns
     for (let j = 0; j < columns; j++) {
-      // b[j]
       const bj = bdata[j][0] || 0
-      // x[j]
       let xj
-      // forward substitution (outer product) avoids inner looping when bj === 0
+
       if (!equalScalar(bj, 0)) {
-        // value @ [j, j]
-        const vjj = data[j][j]
-        // check vjj
+        // non-degenerate row, find solution
+
+        const vjj = mdata[j][j]
+
         if (equalScalar(vjj, 0)) {
-          // system cannot be solved
           throw new Error('Linear system cannot be solved since matrix is singular')
         }
-        // calculate xj
+
         xj = divideScalar(bj, vjj)
+
         // loop rows
         for (let i = j + 1; i < rows; i++) {
-          // update copy of b
-          bdata[i] = [subtract(bdata[i][0] || 0, multiplyScalar(xj, data[i][j]))]
+          bdata[i] = [subtract(bdata[i][0] || 0, multiplyScalar(xj, mdata[i][j]))]
         }
       } else {
-        // zero @ j
+        // degenerate row, we can choose any value
         xj = 0
       }
-      // update x
+
       x[j] = [xj]
     }
-    // return vector
+
     return new DenseMatrix({
       data: x,
       size: [rows, 1]
@@ -112,68 +106,68 @@ export const createLsolve = /* #__PURE__ */ factory(name, dependencies, ({ typed
   function _sparseForwardSubstitution (m, b) {
     // validate matrix and vector, return copy of column vector b
     b = solveValidation(m, b, true)
-    // column vector data
+
     const bdata = b._data
-    // rows & columns
+
     const rows = m._size[0]
     const columns = m._size[1]
-    // matrix arrays
+
     const values = m._values
     const index = m._index
     const ptr = m._ptr
-    // vars
-    let i, k
+
     // result
     const x = []
-    // forward solve m * x = b, loop columns
+
+    // loop columns
     for (let j = 0; j < columns; j++) {
-      // b[j]
       const bj = bdata[j][0] || 0
-      // forward substitution (outer product) avoids inner looping when bj === 0
+
       if (!equalScalar(bj, 0)) {
-        // value @ [j, j]
+        // non-degenerate row, find solution
+
         let vjj = 0
-        // lower triangular matrix values & index (column j)
-        const jvalues = []
-        const jindex = []
-        // last index in column
-        let l = ptr[j + 1]
-        // values in column, find value @ [j, j]
-        for (k = ptr[j]; k < l; k++) {
-          // row
-          i = index[k]
+        // matrix values & indices (column j)
+        const jValues = []
+        const jIndices = []
+
+        // first and last index in the column
+        const firstIndex = ptr[j]
+        const lastIndex = ptr[j + 1]
+
+        // values in column, find value at [j, j]
+        for (let k = firstIndex; k < lastIndex; k++) {
+          const i = index[k]
+
           // check row (rows are not sorted!)
           if (i === j) {
-            // update vjj
             vjj = values[k]
           } else if (i > j) {
             // store lower triangular
-            jvalues.push(values[k])
-            jindex.push(i)
+            jValues.push(values[k])
+            jIndices.push(i)
           }
         }
-        // at this point we must have a value @ [j, j]
+
+        // at this point we must have a value in vjj
         if (equalScalar(vjj, 0)) {
-          // system cannot be solved, there is no value @ [j, j]
           throw new Error('Linear system cannot be solved since matrix is singular')
         }
-        // calculate xj
+
         const xj = divideScalar(bj, vjj)
-        // loop lower triangular
-        for (k = 0, l = jindex.length; k < l; k++) {
-          // row
-          i = jindex[k]
-          // update copy of b
-          bdata[i] = [subtract(bdata[i][0] || 0, multiplyScalar(xj, jvalues[k]))]
+
+        for (let k = 0, l = jIndices.length; k < l; k++) {
+          const i = jIndices[k]
+          bdata[i] = [subtract(bdata[i][0] || 0, multiplyScalar(xj, jValues[k]))]
         }
-        // update x
+
         x[j] = [xj]
       } else {
-        // update x
+        // degenerate row, we can choose any value
         x[j] = [0]
       }
     }
-    // return vector
+
     return new DenseMatrix({
       data: x,
       size: [rows, 1]