feat:node-modules

node_modules/mathjs/lib/esm/function/algebra/solver/lsolve.js

@@ -0,0 +1,157 @@
+import { factory } from '../../../utils/factory.js';
+import { createSolveValidation } from './utils/solveValidation.js';
+var name = 'lsolve';
+var dependencies = ['typed', 'matrix', 'divideScalar', 'multiplyScalar', 'subtractScalar', 'equalScalar', 'DenseMatrix'];
+export var createLsolve = /* #__PURE__ */factory(name, dependencies, _ref => {
+  var {
+    typed,
+    matrix,
+    divideScalar,
+    multiplyScalar,
+    subtractScalar,
+    equalScalar,
+    DenseMatrix
+  } = _ref;
+  var solveValidation = createSolveValidation({
+    DenseMatrix
+  });
+
+  /**
+   * 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`
+   *
+   * Syntax:
+   *
+   *    math.lsolve(L, b)
+   *
+   * Examples:
+   *
+   *    const a = [[-2, 3], [2, 1]]
+   *    const b = [11, 9]
+   *    const x = lsolve(a, b)  // [[-5.5], [20]]
+   *
+   * See also:
+   *
+   *    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
+   *
+   * @return {DenseMatrix | Array}  A column vector with the linear system solution (x)
+   */
+  return typed(name, {
+    'SparseMatrix, Array | Matrix': function SparseMatrix_Array__Matrix(m, b) {
+      return _sparseForwardSubstitution(m, b);
+    },
+    'DenseMatrix, Array | Matrix': function DenseMatrix_Array__Matrix(m, b) {
+      return _denseForwardSubstitution(m, b);
+    },
+    'Array, Array | Matrix': function Array_Array__Matrix(a, b) {
+      var m = matrix(a);
+      var r = _denseForwardSubstitution(m, b);
+      return r.valueOf();
+    }
+  });
+  function _denseForwardSubstitution(m, b) {
+    // validate matrix and vector, return copy of column vector b
+    b = solveValidation(m, b, true);
+    var bdata = b._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+
+    // result
+    var x = [];
+    var mdata = m._data;
+
+    // loop columns
+    for (var j = 0; j < columns; j++) {
+      var bj = bdata[j][0] || 0;
+      var xj = void 0;
+      if (!equalScalar(bj, 0)) {
+        // non-degenerate row, find solution
+
+        var vjj = mdata[j][j];
+        if (equalScalar(vjj, 0)) {
+          throw new Error('Linear system cannot be solved since matrix is singular');
+        }
+        xj = divideScalar(bj, vjj);
+
+        // loop rows
+        for (var i = j + 1; i < rows; i++) {
+          bdata[i] = [subtractScalar(bdata[i][0] || 0, multiplyScalar(xj, mdata[i][j]))];
+        }
+      } else {
+        // degenerate row, we can choose any value
+        xj = 0;
+      }
+      x[j] = [xj];
+    }
+    return new DenseMatrix({
+      data: x,
+      size: [rows, 1]
+    });
+  }
+  function _sparseForwardSubstitution(m, b) {
+    // validate matrix and vector, return copy of column vector b
+    b = solveValidation(m, b, true);
+    var bdata = b._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+    var values = m._values;
+    var index = m._index;
+    var ptr = m._ptr;
+
+    // result
+    var x = [];
+
+    // loop columns
+    for (var j = 0; j < columns; j++) {
+      var bj = bdata[j][0] || 0;
+      if (!equalScalar(bj, 0)) {
+        // non-degenerate row, find solution
+
+        var vjj = 0;
+        // matrix values & indices (column j)
+        var jValues = [];
+        var jIndices = [];
+
+        // first and last index in the column
+        var firstIndex = ptr[j];
+        var lastIndex = ptr[j + 1];
+
+        // values in column, find value at [j, j]
+        for (var k = firstIndex; k < lastIndex; k++) {
+          var i = index[k];
+
+          // check row (rows are not sorted!)
+          if (i === j) {
+            vjj = values[k];
+          } else if (i > j) {
+            // store lower triangular
+            jValues.push(values[k]);
+            jIndices.push(i);
+          }
+        }
+
+        // at this point we must have a value in vjj
+        if (equalScalar(vjj, 0)) {
+          throw new Error('Linear system cannot be solved since matrix is singular');
+        }
+        var xj = divideScalar(bj, vjj);
+        for (var _k = 0, l = jIndices.length; _k < l; _k++) {
+          var _i = jIndices[_k];
+          bdata[_i] = [subtractScalar(bdata[_i][0] || 0, multiplyScalar(xj, jValues[_k]))];
+        }
+        x[j] = [xj];
+      } else {
+        // degenerate row, we can choose any value
+        x[j] = [0];
+      }
+    }
+    return new DenseMatrix({
+      data: x,
+      size: [rows, 1]
+    });
+  }
+});

node_modules/mathjs/lib/esm/function/algebra/solver/lsolveAll.js

@@ -0,0 +1,186 @@
+import { factory } from '../../../utils/factory.js';
+import { createSolveValidation } from './utils/solveValidation.js';
+var name = 'lsolveAll';
+var dependencies = ['typed', 'matrix', 'divideScalar', 'multiplyScalar', 'subtractScalar', 'equalScalar', 'DenseMatrix'];
+export var createLsolveAll = /* #__PURE__ */factory(name, dependencies, _ref => {
+  var {
+    typed,
+    matrix,
+    divideScalar,
+    multiplyScalar,
+    subtractScalar,
+    equalScalar,
+    DenseMatrix
+  } = _ref;
+  var solveValidation = createSolveValidation({
+    DenseMatrix
+  });
+
+  /**
+   * Finds all solutions of a linear equation system by forwards substitution. Matrix must be a lower triangular matrix.
+   *
+   * `L * x = b`
+   *
+   * Syntax:
+   *
+   *    math.lsolveAll(L, b)
+   *
+   * Examples:
+   *
+   *    const a = [[-2, 3], [2, 1]]
+   *    const b = [11, 9]
+   *    const x = lsolveAll(a, b)  // [ [[-5.5], [20]] ]
+   *
+   * See also:
+   *
+   *    lsolve, 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
+   *
+   * @return {DenseMatrix[] | Array[]}  An array of affine-independent column vectors (x) that solve the linear system
+   */
+  return typed(name, {
+    'SparseMatrix, Array | Matrix': function SparseMatrix_Array__Matrix(m, b) {
+      return _sparseForwardSubstitution(m, b);
+    },
+    'DenseMatrix, Array | Matrix': function DenseMatrix_Array__Matrix(m, b) {
+      return _denseForwardSubstitution(m, b);
+    },
+    'Array, Array | Matrix': function Array_Array__Matrix(a, b) {
+      var m = matrix(a);
+      var R = _denseForwardSubstitution(m, b);
+      return R.map(r => r.valueOf());
+    }
+  });
+  function _denseForwardSubstitution(m, b_) {
+    // the algorithm is derived from
+    // https://www.overleaf.com/read/csvgqdxggyjv
+
+    // array of right-hand sides
+    var B = [solveValidation(m, b_, true)._data.map(e => e[0])];
+    var M = m._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+
+    // loop columns
+    for (var i = 0; i < columns; i++) {
+      var L = B.length;
+
+      // loop right-hand sides
+      for (var k = 0; k < L; k++) {
+        var b = B[k];
+        if (!equalScalar(M[i][i], 0)) {
+          // non-singular row
+
+          b[i] = divideScalar(b[i], M[i][i]);
+          for (var j = i + 1; j < columns; j++) {
+            // b[j] -= b[i] * M[j,i]
+            b[j] = subtractScalar(b[j], multiplyScalar(b[i], M[j][i]));
+          }
+        } else if (!equalScalar(b[i], 0)) {
+          // singular row, nonzero RHS
+
+          if (k === 0) {
+            // There is no valid solution
+            return [];
+          } else {
+            // This RHS is invalid but other solutions may still exist
+            B.splice(k, 1);
+            k -= 1;
+            L -= 1;
+          }
+        } else if (k === 0) {
+          // singular row, RHS is zero
+
+          var bNew = [...b];
+          bNew[i] = 1;
+          for (var _j = i + 1; _j < columns; _j++) {
+            bNew[_j] = subtractScalar(bNew[_j], M[_j][i]);
+          }
+          B.push(bNew);
+        }
+      }
+    }
+    return B.map(x => new DenseMatrix({
+      data: x.map(e => [e]),
+      size: [rows, 1]
+    }));
+  }
+  function _sparseForwardSubstitution(m, b_) {
+    // array of right-hand sides
+    var B = [solveValidation(m, b_, true)._data.map(e => e[0])];
+    var rows = m._size[0];
+    var columns = m._size[1];
+    var values = m._values;
+    var index = m._index;
+    var ptr = m._ptr;
+
+    // loop columns
+    for (var i = 0; i < columns; i++) {
+      var L = B.length;
+
+      // loop right-hand sides
+      for (var k = 0; k < L; k++) {
+        var b = B[k];
+
+        // values & indices (column i)
+        var iValues = [];
+        var iIndices = [];
+
+        // first & last indeces in column
+        var firstIndex = ptr[i];
+        var lastIndex = ptr[i + 1];
+
+        // find the value at [i, i]
+        var Mii = 0;
+        for (var j = firstIndex; j < lastIndex; j++) {
+          var J = index[j];
+          // check row
+          if (J === i) {
+            Mii = values[j];
+          } else if (J > i) {
+            // store lower triangular
+            iValues.push(values[j]);
+            iIndices.push(J);
+          }
+        }
+        if (!equalScalar(Mii, 0)) {
+          // non-singular row
+
+          b[i] = divideScalar(b[i], Mii);
+          for (var _j2 = 0, _lastIndex = iIndices.length; _j2 < _lastIndex; _j2++) {
+            var _J = iIndices[_j2];
+            b[_J] = subtractScalar(b[_J], multiplyScalar(b[i], iValues[_j2]));
+          }
+        } else if (!equalScalar(b[i], 0)) {
+          // singular row, nonzero RHS
+
+          if (k === 0) {
+            // There is no valid solution
+            return [];
+          } else {
+            // This RHS is invalid but other solutions may still exist
+            B.splice(k, 1);
+            k -= 1;
+            L -= 1;
+          }
+        } else if (k === 0) {
+          // singular row, RHS is zero
+
+          var bNew = [...b];
+          bNew[i] = 1;
+          for (var _j3 = 0, _lastIndex2 = iIndices.length; _j3 < _lastIndex2; _j3++) {
+            var _J2 = iIndices[_j3];
+            bNew[_J2] = subtractScalar(bNew[_J2], iValues[_j3]);
+          }
+          B.push(bNew);
+        }
+      }
+    }
+    return B.map(x => new DenseMatrix({
+      data: x.map(e => [e]),
+      size: [rows, 1]
+    }));
+  }
+});

node_modules/mathjs/lib/esm/function/algebra/solver/lusolve.js

@@ -0,0 +1,108 @@
+import { isArray, isMatrix } from '../../../utils/is.js';
+import { factory } from '../../../utils/factory.js';
+import { createSolveValidation } from './utils/solveValidation.js';
+import { csIpvec } from '../sparse/csIpvec.js';
+var name = 'lusolve';
+var dependencies = ['typed', 'matrix', 'lup', 'slu', 'usolve', 'lsolve', 'DenseMatrix'];
+export var createLusolve = /* #__PURE__ */factory(name, dependencies, _ref => {
+  var {
+    typed,
+    matrix,
+    lup,
+    slu,
+    usolve,
+    lsolve,
+    DenseMatrix
+  } = _ref;
+  var solveValidation = createSolveValidation({
+    DenseMatrix
+  });
+
+  /**
+   * Solves the linear system `A * x = b` where `A` is an [n x n] matrix and `b` is a [n] column vector.
+   *
+   * Syntax:
+   *
+   *    math.lusolve(A, b)     // returns column vector with the solution to the linear system A * x = b
+   *    math.lusolve(lup, b)   // returns column vector with the solution to the linear system A * x = b, lup = math.lup(A)
+   *
+   * Examples:
+   *
+   *    const m = [[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 3, 0], [0, 0, 0, 4]]
+   *
+   *    const x = math.lusolve(m, [-1, -1, -1, -1])        // x = [[-1], [-0.5], [-1/3], [-0.25]]
+   *
+   *    const f = math.lup(m)
+   *    const x1 = math.lusolve(f, [-1, -1, -1, -1])       // x1 = [[-1], [-0.5], [-1/3], [-0.25]]
+   *    const x2 = math.lusolve(f, [1, 2, 1, -1])          // x2 = [[1], [1], [1/3], [-0.25]]
+   *
+   *    const a = [[-2, 3], [2, 1]]
+   *    const b = [11, 9]
+   *    const x = math.lusolve(a, b)  // [[2], [5]]
+   *
+   * See also:
+   *
+   *    lup, slu, lsolve, usolve
+   *
+   * @param {Matrix | Array | Object} A      Invertible Matrix or the Matrix LU decomposition
+   * @param {Matrix | Array} b               Column Vector
+   * @param {number} [order]                 The Symbolic Ordering and Analysis order, see slu for details. Matrix must be a SparseMatrix
+   * @param {Number} [threshold]             Partial pivoting threshold (1 for partial pivoting), see slu for details. Matrix must be a SparseMatrix.
+   *
+   * @return {DenseMatrix | Array}           Column vector with the solution to the linear system A * x = b
+   */
+  return typed(name, {
+    'Array, Array | Matrix': function Array_Array__Matrix(a, b) {
+      a = matrix(a);
+      var d = lup(a);
+      var x = _lusolve(d.L, d.U, d.p, null, b);
+      return x.valueOf();
+    },
+    'DenseMatrix, Array | Matrix': function DenseMatrix_Array__Matrix(a, b) {
+      var d = lup(a);
+      return _lusolve(d.L, d.U, d.p, null, b);
+    },
+    'SparseMatrix, Array | Matrix': function SparseMatrix_Array__Matrix(a, b) {
+      var d = lup(a);
+      return _lusolve(d.L, d.U, d.p, null, b);
+    },
+    'SparseMatrix, Array | Matrix, number, number': function SparseMatrix_Array__Matrix_number_number(a, b, order, threshold) {
+      var d = slu(a, order, threshold);
+      return _lusolve(d.L, d.U, d.p, d.q, b);
+    },
+    'Object, Array | Matrix': function Object_Array__Matrix(d, b) {
+      return _lusolve(d.L, d.U, d.p, d.q, b);
+    }
+  });
+  function _toMatrix(a) {
+    if (isMatrix(a)) {
+      return a;
+    }
+    if (isArray(a)) {
+      return matrix(a);
+    }
+    throw new TypeError('Invalid Matrix LU decomposition');
+  }
+  function _lusolve(l, u, p, q, b) {
+    // verify decomposition
+    l = _toMatrix(l);
+    u = _toMatrix(u);
+
+    // apply row permutations if needed (b is a DenseMatrix)
+    if (p) {
+      b = solveValidation(l, b, true);
+      b._data = csIpvec(p, b._data);
+    }
+
+    // use forward substitution to resolve L * y = b
+    var y = lsolve(l, b);
+    // use backward substitution to resolve U * x = y
+    var x = usolve(u, y);
+
+    // apply column permutations if needed (x is a DenseMatrix)
+    if (q) {
+      x._data = csIpvec(q, x._data);
+    }
+    return x;
+  }
+});

node_modules/mathjs/lib/esm/function/algebra/solver/usolve.js

@@ -0,0 +1,161 @@
+import { factory } from '../../../utils/factory.js';
+import { createSolveValidation } from './utils/solveValidation.js';
+var name = 'usolve';
+var dependencies = ['typed', 'matrix', 'divideScalar', 'multiplyScalar', 'subtractScalar', 'equalScalar', 'DenseMatrix'];
+export var createUsolve = /* #__PURE__ */factory(name, dependencies, _ref => {
+  var {
+    typed,
+    matrix,
+    divideScalar,
+    multiplyScalar,
+    subtractScalar,
+    equalScalar,
+    DenseMatrix
+  } = _ref;
+  var solveValidation = createSolveValidation({
+    DenseMatrix
+  });
+
+  /**
+   * Finds one solution of a linear equation system by backward substitution. Matrix must be an upper triangular matrix. Throws an error if there's no solution.
+   *
+   * `U * x = b`
+   *
+   * Syntax:
+   *
+   *    math.usolve(U, b)
+   *
+   * Examples:
+   *
+   *    const a = [[-2, 3], [2, 1]]
+   *    const b = [11, 9]
+   *    const x = usolve(a, b)  // [[8], [9]]
+   *
+   * See also:
+   *
+   *    usolveAll, lup, slu, usolve, lusolve
+   *
+   * @param {Matrix, Array} U       A N x N matrix or array (U)
+   * @param {Matrix, Array} b       A column vector with the b values
+   *
+   * @return {DenseMatrix | Array}  A column vector with the linear system solution (x)
+   */
+  return typed(name, {
+    'SparseMatrix, Array | Matrix': function SparseMatrix_Array__Matrix(m, b) {
+      return _sparseBackwardSubstitution(m, b);
+    },
+    'DenseMatrix, Array | Matrix': function DenseMatrix_Array__Matrix(m, b) {
+      return _denseBackwardSubstitution(m, b);
+    },
+    'Array, Array | Matrix': function Array_Array__Matrix(a, b) {
+      var m = matrix(a);
+      var r = _denseBackwardSubstitution(m, b);
+      return r.valueOf();
+    }
+  });
+  function _denseBackwardSubstitution(m, b) {
+    // make b into a column vector
+    b = solveValidation(m, b, true);
+    var bdata = b._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+
+    // result
+    var x = [];
+    var mdata = m._data;
+    // loop columns backwards
+    for (var j = columns - 1; j >= 0; j--) {
+      // b[j]
+      var bj = bdata[j][0] || 0;
+      // x[j]
+      var xj = void 0;
+      if (!equalScalar(bj, 0)) {
+        // value at [j, j]
+        var vjj = mdata[j][j];
+        if (equalScalar(vjj, 0)) {
+          // system cannot be solved
+          throw new Error('Linear system cannot be solved since matrix is singular');
+        }
+        xj = divideScalar(bj, vjj);
+
+        // loop rows
+        for (var i = j - 1; i >= 0; i--) {
+          // update copy of b
+          bdata[i] = [subtractScalar(bdata[i][0] || 0, multiplyScalar(xj, mdata[i][j]))];
+        }
+      } else {
+        // zero value at j
+        xj = 0;
+      }
+      // update x
+      x[j] = [xj];
+    }
+    return new DenseMatrix({
+      data: x,
+      size: [rows, 1]
+    });
+  }
+  function _sparseBackwardSubstitution(m, b) {
+    // make b into a column vector
+    b = solveValidation(m, b, true);
+    var bdata = b._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+    var values = m._values;
+    var index = m._index;
+    var ptr = m._ptr;
+
+    // result
+    var x = [];
+
+    // loop columns backwards
+    for (var j = columns - 1; j >= 0; j--) {
+      var bj = bdata[j][0] || 0;
+      if (!equalScalar(bj, 0)) {
+        // non-degenerate row, find solution
+
+        var vjj = 0;
+
+        // upper triangular matrix values & index (column j)
+        var jValues = [];
+        var jIndices = [];
+
+        // first & last indeces in column
+        var firstIndex = ptr[j];
+        var lastIndex = ptr[j + 1];
+
+        // values in column, find value at [j, j], loop backwards
+        for (var k = lastIndex - 1; k >= firstIndex; k--) {
+          var i = index[k];
+
+          // check row (rows are not sorted!)
+          if (i === j) {
+            vjj = values[k];
+          } else if (i < j) {
+            // store upper triangular
+            jValues.push(values[k]);
+            jIndices.push(i);
+          }
+        }
+
+        // at this point we must have a value in vjj
+        if (equalScalar(vjj, 0)) {
+          throw new Error('Linear system cannot be solved since matrix is singular');
+        }
+        var xj = divideScalar(bj, vjj);
+        for (var _k = 0, _lastIndex = jIndices.length; _k < _lastIndex; _k++) {
+          var _i = jIndices[_k];
+          bdata[_i] = [subtractScalar(bdata[_i][0], multiplyScalar(xj, jValues[_k]))];
+        }
+        x[j] = [xj];
+      } else {
+        // degenerate row, we can choose any value
+        x[j] = [0];
+      }
+    }
+    return new DenseMatrix({
+      data: x,
+      size: [rows, 1]
+    });
+  }
+});

node_modules/mathjs/lib/esm/function/algebra/solver/usolveAll.js

@@ -0,0 +1,190 @@
+import { factory } from '../../../utils/factory.js';
+import { createSolveValidation } from './utils/solveValidation.js';
+var name = 'usolveAll';
+var dependencies = ['typed', 'matrix', 'divideScalar', 'multiplyScalar', 'subtractScalar', 'equalScalar', 'DenseMatrix'];
+export var createUsolveAll = /* #__PURE__ */factory(name, dependencies, _ref => {
+  var {
+    typed,
+    matrix,
+    divideScalar,
+    multiplyScalar,
+    subtractScalar,
+    equalScalar,
+    DenseMatrix
+  } = _ref;
+  var solveValidation = createSolveValidation({
+    DenseMatrix
+  });
+
+  /**
+   * Finds all solutions of a linear equation system by backward substitution. Matrix must be an upper triangular matrix.
+   *
+   * `U * x = b`
+   *
+   * Syntax:
+   *
+   *    math.usolveAll(U, b)
+   *
+   * Examples:
+   *
+   *    const a = [[-2, 3], [2, 1]]
+   *    const b = [11, 9]
+   *    const x = usolveAll(a, b)  // [ [[8], [9]] ]
+   *
+   * See also:
+   *
+   *    usolve, lup, slu, usolve, lusolve
+   *
+   * @param {Matrix, Array} U       A N x N matrix or array (U)
+   * @param {Matrix, Array} b       A column vector with the b values
+   *
+   * @return {DenseMatrix[] | Array[]}  An array of affine-independent column vectors (x) that solve the linear system
+   */
+  return typed(name, {
+    'SparseMatrix, Array | Matrix': function SparseMatrix_Array__Matrix(m, b) {
+      return _sparseBackwardSubstitution(m, b);
+    },
+    'DenseMatrix, Array | Matrix': function DenseMatrix_Array__Matrix(m, b) {
+      return _denseBackwardSubstitution(m, b);
+    },
+    'Array, Array | Matrix': function Array_Array__Matrix(a, b) {
+      var m = matrix(a);
+      var R = _denseBackwardSubstitution(m, b);
+      return R.map(r => r.valueOf());
+    }
+  });
+  function _denseBackwardSubstitution(m, b_) {
+    // the algorithm is derived from
+    // https://www.overleaf.com/read/csvgqdxggyjv
+
+    // array of right-hand sides
+    var B = [solveValidation(m, b_, true)._data.map(e => e[0])];
+    var M = m._data;
+    var rows = m._size[0];
+    var columns = m._size[1];
+
+    // loop columns backwards
+    for (var i = columns - 1; i >= 0; i--) {
+      var L = B.length;
+
+      // loop right-hand sides
+      for (var k = 0; k < L; k++) {
+        var b = B[k];
+        if (!equalScalar(M[i][i], 0)) {
+          // non-singular row
+
+          b[i] = divideScalar(b[i], M[i][i]);
+          for (var j = i - 1; j >= 0; j--) {
+            // b[j] -= b[i] * M[j,i]
+            b[j] = subtractScalar(b[j], multiplyScalar(b[i], M[j][i]));
+          }
+        } else if (!equalScalar(b[i], 0)) {
+          // singular row, nonzero RHS
+
+          if (k === 0) {
+            // There is no valid solution
+            return [];
+          } else {
+            // This RHS is invalid but other solutions may still exist
+            B.splice(k, 1);
+            k -= 1;
+            L -= 1;
+          }
+        } else if (k === 0) {
+          // singular row, RHS is zero
+
+          var bNew = [...b];
+          bNew[i] = 1;
+          for (var _j = i - 1; _j >= 0; _j--) {
+            bNew[_j] = subtractScalar(bNew[_j], M[_j][i]);
+          }
+          B.push(bNew);
+        }
+      }
+    }
+    return B.map(x => new DenseMatrix({
+      data: x.map(e => [e]),
+      size: [rows, 1]
+    }));
+  }
+  function _sparseBackwardSubstitution(m, b_) {
+    // array of right-hand sides
+    var B = [solveValidation(m, b_, true)._data.map(e => e[0])];
+    var rows = m._size[0];
+    var columns = m._size[1];
+    var values = m._values;
+    var index = m._index;
+    var ptr = m._ptr;
+
+    // loop columns backwards
+    for (var i = columns - 1; i >= 0; i--) {
+      var L = B.length;
+
+      // loop right-hand sides
+      for (var k = 0; k < L; k++) {
+        var b = B[k];
+
+        // values & indices (column i)
+        var iValues = [];
+        var iIndices = [];
+
+        // first & last indeces in column
+        var firstIndex = ptr[i];
+        var lastIndex = ptr[i + 1];
+
+        // find the value at [i, i]
+        var Mii = 0;
+        for (var j = lastIndex - 1; j >= firstIndex; j--) {
+          var J = index[j];
+          // check row
+          if (J === i) {
+            Mii = values[j];
+          } else if (J < i) {
+            // store upper triangular
+            iValues.push(values[j]);
+            iIndices.push(J);
+          }
+        }
+        if (!equalScalar(Mii, 0)) {
+          // non-singular row
+
+          b[i] = divideScalar(b[i], Mii);
+
+          // loop upper triangular
+          for (var _j2 = 0, _lastIndex = iIndices.length; _j2 < _lastIndex; _j2++) {
+            var _J = iIndices[_j2];
+            b[_J] = subtractScalar(b[_J], multiplyScalar(b[i], iValues[_j2]));
+          }
+        } else if (!equalScalar(b[i], 0)) {
+          // singular row, nonzero RHS
+
+          if (k === 0) {
+            // There is no valid solution
+            return [];
+          } else {
+            // This RHS is invalid but other solutions may still exist
+            B.splice(k, 1);
+            k -= 1;
+            L -= 1;
+          }
+        } else if (k === 0) {
+          // singular row, RHS is zero
+
+          var bNew = [...b];
+          bNew[i] = 1;
+
+          // loop upper triangular
+          for (var _j3 = 0, _lastIndex2 = iIndices.length; _j3 < _lastIndex2; _j3++) {
+            var _J2 = iIndices[_j3];
+            bNew[_J2] = subtractScalar(bNew[_J2], iValues[_j3]);
+          }
+          B.push(bNew);
+        }
+      }
+    }
+    return B.map(x => new DenseMatrix({
+      data: x.map(e => [e]),
+      size: [rows, 1]
+    }));
+  }
+});

node_modules/mathjs/lib/esm/function/algebra/solver/utils/solveValidation.js

@@ -0,0 +1,115 @@
+import { isArray, isMatrix, isDenseMatrix, isSparseMatrix } from '../../../../utils/is.js';
+import { arraySize } from '../../../../utils/array.js';
+import { format } from '../../../../utils/string.js';
+export function createSolveValidation(_ref) {
+  var {
+    DenseMatrix
+  } = _ref;
+  /**
+   * Validates matrix and column vector b for backward/forward substitution algorithms.
+   *
+   * @param {Matrix} m            An N x N matrix
+   * @param {Array | Matrix} b    A column vector
+   * @param {Boolean} copy        Return a copy of vector b
+   *
+   * @return {DenseMatrix}        Dense column vector b
+   */
+  return function solveValidation(m, b, copy) {
+    var mSize = m.size();
+    if (mSize.length !== 2) {
+      throw new RangeError('Matrix must be two dimensional (size: ' + format(mSize) + ')');
+    }
+    var rows = mSize[0];
+    var columns = mSize[1];
+    if (rows !== columns) {
+      throw new RangeError('Matrix must be square (size: ' + format(mSize) + ')');
+    }
+    var data = [];
+    if (isMatrix(b)) {
+      var bSize = b.size();
+      var bdata = b._data;
+
+      // 1-dim vector
+      if (bSize.length === 1) {
+        if (bSize[0] !== rows) {
+          throw new RangeError('Dimension mismatch. Matrix columns must match vector length.');
+        }
+        for (var i = 0; i < rows; i++) {
+          data[i] = [bdata[i]];
+        }
+        return new DenseMatrix({
+          data,
+          size: [rows, 1],
+          datatype: b._datatype
+        });
+      }
+
+      // 2-dim column
+      if (bSize.length === 2) {
+        if (bSize[0] !== rows || bSize[1] !== 1) {
+          throw new RangeError('Dimension mismatch. Matrix columns must match vector length.');
+        }
+        if (isDenseMatrix(b)) {
+          if (copy) {
+            data = [];
+            for (var _i = 0; _i < rows; _i++) {
+              data[_i] = [bdata[_i][0]];
+            }
+            return new DenseMatrix({
+              data,
+              size: [rows, 1],
+              datatype: b._datatype
+            });
+          }
+          return b;
+        }
+        if (isSparseMatrix(b)) {
+          for (var _i2 = 0; _i2 < rows; _i2++) {
+            data[_i2] = [0];
+          }
+          var values = b._values;
+          var index = b._index;
+          var ptr = b._ptr;
+          for (var k1 = ptr[1], k = ptr[0]; k < k1; k++) {
+            var _i3 = index[k];
+            data[_i3][0] = values[k];
+          }
+          return new DenseMatrix({
+            data,
+            size: [rows, 1],
+            datatype: b._datatype
+          });
+        }
+      }
+      throw new RangeError('Dimension mismatch. The right side has to be either 1- or 2-dimensional vector.');
+    }
+    if (isArray(b)) {
+      var bsize = arraySize(b);
+      if (bsize.length === 1) {
+        if (bsize[0] !== rows) {
+          throw new RangeError('Dimension mismatch. Matrix columns must match vector length.');
+        }
+        for (var _i4 = 0; _i4 < rows; _i4++) {
+          data[_i4] = [b[_i4]];
+        }
+        return new DenseMatrix({
+          data,
+          size: [rows, 1]
+        });
+      }
+      if (bsize.length === 2) {
+        if (bsize[0] !== rows || bsize[1] !== 1) {
+          throw new RangeError('Dimension mismatch. Matrix columns must match vector length.');
+        }
+        for (var _i5 = 0; _i5 < rows; _i5++) {
+          data[_i5] = [b[_i5][0]];
+        }
+        return new DenseMatrix({
+          data,
+          size: [rows, 1]
+        });
+      }
+      throw new RangeError('Dimension mismatch. The right side has to be either 1- or 2-dimensional vector.');
+    }
+  };
+}