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source: branches/2994-AutoDiffForIntervals/HeuristicLab.Problems.DataAnalysis.Symbolic/3.4/Interpreter/Interpreter.cs @ 17131

Last change on this file since 17131 was 17131, checked in by gkronber, 3 years ago

#2994: added support for tanh symbol in new interpreters

File size: 51.5 KB
Line 
1using System;
2using System.Collections.Generic;
3using System.Diagnostics;
4using System.Linq;
5using HeuristicLab.Common;
6using HeuristicLab.Encodings.SymbolicExpressionTreeEncoding;
7
8namespace HeuristicLab.Problems.DataAnalysis.Symbolic {
9  public abstract class Interpreter<T> where T : IAlgebraicType<T> {
10    public struct Instruction {
11      public byte opcode;
12      public ushort narg;
13      public int childIndex;
14      public double dblVal;
15      public object data; // any kind of data you want to store in instructions
16      public T value;
17    }
18
19    public T Evaluate(Instruction[] code) {
20      for (int i = code.Length - 1; i >= 0; --i) {
21        var instr = code[i];
22        var c = instr.childIndex;
23        var n = instr.narg;
24
25        switch (instr.opcode) {
26          case OpCodes.Variable: {
27              LoadVariable(instr);
28              break;
29            }
30          case OpCodes.Constant: { break; }  // we initialize constants in Compile. The value never changes afterwards
31          case OpCodes.Add: {
32              instr.value.Assign(code[c].value);
33              for (int j = 1; j < n; ++j) {
34                instr.value.Add(code[c + j].value);
35              }
36              break;
37            }
38
39          case OpCodes.Sub: {
40              if (n == 1) {
41                instr.value.AssignNeg(code[c].value);
42              } else {
43                instr.value.Assign(code[c].value);
44                for (int j = 1; j < n; ++j) {
45                  instr.value.Sub(code[c + j].value);
46                }
47              }
48              break;
49            }
50
51          case OpCodes.Mul: {
52              instr.value.Assign(code[c].value);
53              for (int j = 1; j < n; ++j) {
54                instr.value.Mul(code[c + j].value);
55              }
56              break;
57            }
58
59          case OpCodes.Div: {
60              if (n == 1) {
61                instr.value.AssignInv(code[c].value);
62              } else {
63                instr.value.Assign(code[c].value);
64                for (int j = 1; j < n; ++j) {
65                  instr.value.Div(code[c + j].value);
66                }
67              }
68              break;
69            }
70          case OpCodes.Square: {
71              instr.value.AssignIntPower(code[c].value, 2);
72              break;
73            }
74          case OpCodes.SquareRoot: {
75              instr.value.AssignIntRoot(code[c].value, 2);
76              break;
77            }
78          case OpCodes.Cube: {
79              instr.value.AssignIntPower(code[c].value, 3);
80              break;
81            }
82          case OpCodes.CubeRoot: {
83              instr.value.AssignIntRoot(code[c].value, 3);
84              break;
85            }
86          case OpCodes.Exp: {
87              instr.value.AssignExp(code[c].value);
88              break;
89            }
90          case OpCodes.Log: {
91              instr.value.AssignLog(code[c].value);
92              break;
93            }
94          case OpCodes.Sin: {
95              instr.value.AssignSin(code[c].value);
96              break;
97            }
98          case OpCodes.Cos: {
99              instr.value.AssignCos(code[c].value);
100              break;
101            }
102          case OpCodes.Absolute: {
103              instr.value.AssignAbs(code[c].value);
104              break;
105            }
106          case OpCodes.AnalyticQuotient: {
107              instr.value.Assign(code[c].value);
108              for (int j = 1; j < n; ++j) {
109                var t = instr.value.One;
110                t.Add(code[c + j].value.Clone().IntPower(2));
111                instr.value.Div(t.IntRoot(2));
112              }
113              break;
114            }
115          case OpCodes.Tanh: {
116              instr.value.AssignTanh(code[c].value);
117              break;
118            }
119
120          default: throw new ArgumentException($"Unknown opcode {instr.opcode}");
121        }
122      }
123      return code[0].value;
124    }
125
126    protected Instruction[] Compile(ISymbolicExpressionTree tree) {
127      var root = tree.Root.GetSubtree(0).GetSubtree(0);
128      var code = new Instruction[root.GetLength()];
129      if (root.SubtreeCount > ushort.MaxValue) throw new ArgumentException("Number of subtrees is too big (>65.535)");
130      int c = 1, i = 0;
131      foreach (var node in root.IterateNodesBreadth()) {
132        if (node.SubtreeCount > ushort.MaxValue) throw new ArgumentException("Number of subtrees is too big (>65.535)");
133        code[i] = new Instruction {
134          opcode = OpCodes.MapSymbolToOpCode(node),
135          narg = (ushort)node.SubtreeCount,
136          childIndex = c
137        };
138        if (node is VariableTreeNode variable) {
139          InitializeTerminalInstruction(ref code[i], variable);
140        } else if (node is ConstantTreeNode constant) {
141          InitializeTerminalInstruction(ref code[i], constant);
142        } else {
143          InitializeInternalInstruction(ref code[i], node);
144        }
145        c += node.SubtreeCount;
146        ++i;
147      }
148      return code;
149    }
150
151    protected abstract void InitializeTerminalInstruction(ref Instruction instruction, ConstantTreeNode constant);
152    protected abstract void InitializeTerminalInstruction(ref Instruction instruction, VariableTreeNode variable);
153    protected abstract void InitializeInternalInstruction(ref Instruction instruction, ISymbolicExpressionTreeNode node);
154
155    protected abstract void LoadVariable(Instruction a);
156
157  }
158
159
160  public sealed class VectorEvaluator : Interpreter<AlgebraicDoubleVector> {
161    private const int BATCHSIZE = 128;
162    [ThreadStatic]
163    private Dictionary<string, double[]> cachedData;
164
165    [ThreadStatic]
166    private IDataset dataset;
167
168    [ThreadStatic]
169    private int rowIndex;
170
171    [ThreadStatic]
172    private int[] rows;
173
174    private void InitCache(IDataset dataset) {
175      this.dataset = dataset;
176      cachedData = new Dictionary<string, double[]>();
177      foreach (var v in dataset.DoubleVariables) {
178        cachedData[v] = dataset.GetReadOnlyDoubleValues(v).ToArray();
179      }
180    }
181
182    public double[] Evaluate(ISymbolicExpressionTree tree, IDataset dataset, int[] rows) {
183      if (cachedData == null || this.dataset != dataset) {
184        InitCache(dataset);
185      }
186
187      this.rows = rows;
188      var code = Compile(tree);
189      var remainingRows = rows.Length % BATCHSIZE;
190      var roundedTotal = rows.Length - remainingRows;
191
192      var result = new double[rows.Length];
193
194      for (rowIndex = 0; rowIndex < roundedTotal; rowIndex += BATCHSIZE) {
195        Evaluate(code);
196        code[0].value.CopyTo(result, rowIndex, BATCHSIZE);
197      }
198
199      if (remainingRows > 0) {
200        Evaluate(code);
201        code[0].value.CopyTo(result, roundedTotal, remainingRows);
202      }
203
204      return result;
205    }
206
207    protected override void InitializeTerminalInstruction(ref Instruction instruction, ConstantTreeNode constant) {
208      instruction.dblVal = constant.Value;
209      instruction.value = new AlgebraicDoubleVector(BATCHSIZE);
210      instruction.value.AssignConstant(instruction.dblVal);
211    }
212
213    protected override void InitializeTerminalInstruction(ref Instruction instruction, VariableTreeNode variable) {
214      instruction.dblVal = variable.Weight;
215      instruction.value = new AlgebraicDoubleVector(BATCHSIZE);
216      if (cachedData.ContainsKey(variable.VariableName)) {
217        instruction.data = cachedData[variable.VariableName];
218      } else {
219        instruction.data = dataset.GetDoubleValues(variable.VariableName).ToArray();
220        cachedData[variable.VariableName] = (double[])instruction.data;
221      }
222    }
223
224    protected override void InitializeInternalInstruction(ref Instruction instruction, ISymbolicExpressionTreeNode node) {
225      instruction.value = new AlgebraicDoubleVector(BATCHSIZE);
226    }
227
228    protected override void LoadVariable(Instruction a) {
229      var data = (double[])a.data;
230      for (int i = rowIndex; i < rows.Length && i - rowIndex < BATCHSIZE; i++) a.value[i - rowIndex] = data[rows[i]];
231      a.value.Scale(a.dblVal);
232    }
233  }
234
235  public sealed class VectorAutoDiffEvaluator : Interpreter<MultivariateDual<AlgebraicDoubleVector>> {
236    private const int BATCHSIZE = 128;
237    [ThreadStatic]
238    private Dictionary<string, double[]> cachedData;
239
240    [ThreadStatic]
241    private IDataset dataset;
242
243    [ThreadStatic]
244    private int rowIndex;
245
246    [ThreadStatic]
247    private int[] rows;
248
249    [ThreadStatic]
250    private Dictionary<ISymbolicExpressionTreeNode, int> node2paramIdx;
251
252    private void InitCache(IDataset dataset) {
253      this.dataset = dataset;
254      cachedData = new Dictionary<string, double[]>();
255      foreach (var v in dataset.DoubleVariables) {
256        cachedData[v] = dataset.GetDoubleValues(v).ToArray();
257      }
258    }
259
260    /// <summary>
261    ///
262    /// </summary>
263    /// <param name="tree"></param>
264    /// <param name="dataset"></param>
265    /// <param name="rows"></param>
266    /// <param name="parameterNodes"></param>
267    /// <param name="fi">Function output. Must be allocated by the caller.</param>
268    /// <param name="jac">Jacobian matrix. Must be allocated by the caller.</param>
269    public void Evaluate(ISymbolicExpressionTree tree, IDataset dataset, int[] rows, ISymbolicExpressionTreeNode[] parameterNodes, double[] fi, double[,] jac) {
270      if (cachedData == null || this.dataset != dataset) {
271        InitCache(dataset);
272      }
273
274      int nParams = parameterNodes.Length;
275      node2paramIdx = new Dictionary<ISymbolicExpressionTreeNode, int>();
276      for (int i = 0; i < parameterNodes.Length; i++) node2paramIdx.Add(parameterNodes[i], i);
277
278      var code = Compile(tree);
279
280      var remainingRows = rows.Length % BATCHSIZE;
281      var roundedTotal = rows.Length - remainingRows;
282
283      this.rows = rows;
284
285      for (rowIndex = 0; rowIndex < roundedTotal; rowIndex += BATCHSIZE) {
286        Evaluate(code);
287        code[0].value.Value.CopyTo(fi, rowIndex, BATCHSIZE);
288
289        // TRANSPOSE into JAC
290        var g = code[0].value.Gradient;
291        for (int j = 0; j < nParams; ++j) {
292          if (g.Elements.TryGetValue(j, out AlgebraicDoubleVector v)) {
293            v.CopyColumnTo(jac, j, rowIndex, BATCHSIZE);
294          } else {
295            for (int r = 0; r < BATCHSIZE; r++) jac[rowIndex + r, j] = 0.0;
296          }
297        }
298      }
299
300      if (remainingRows > 0) {
301        Evaluate(code);
302        code[0].value.Value.CopyTo(fi, roundedTotal, remainingRows);
303
304        var g = code[0].value.Gradient;
305        for (int j = 0; j < nParams; ++j)
306          if (g.Elements.TryGetValue(j, out AlgebraicDoubleVector v)) {
307            v.CopyColumnTo(jac, j, roundedTotal, remainingRows);
308          } else {
309            for (int r = 0; r < remainingRows; r++) jac[roundedTotal + r, j] = 0.0;
310          }
311      }
312    }
313
314    protected override void InitializeInternalInstruction(ref Instruction instruction, ISymbolicExpressionTreeNode node) {
315      var zero = new AlgebraicDoubleVector(BATCHSIZE);
316      instruction.value = new MultivariateDual<AlgebraicDoubleVector>(zero);
317    }
318
319    protected override void InitializeTerminalInstruction(ref Instruction instruction, ConstantTreeNode constant) {
320      var g_arr = new double[BATCHSIZE];
321      if (node2paramIdx.TryGetValue(constant, out var paramIdx)) {
322        for (int i = 0; i < BATCHSIZE; i++) g_arr[i] = 1.0;
323        var g = new AlgebraicDoubleVector(g_arr);
324        instruction.value = new MultivariateDual<AlgebraicDoubleVector>(new AlgebraicDoubleVector(BATCHSIZE), paramIdx, g); // only a single column for the gradient
325      } else {
326        instruction.value = new MultivariateDual<AlgebraicDoubleVector>(new AlgebraicDoubleVector(BATCHSIZE));
327      }
328
329      instruction.dblVal = constant.Value;
330      instruction.value.Value.AssignConstant(instruction.dblVal);
331    }
332
333    protected override void InitializeTerminalInstruction(ref Instruction instruction, VariableTreeNode variable) {
334      double[] data;
335      if (cachedData.ContainsKey(variable.VariableName)) {
336        data = cachedData[variable.VariableName];
337      } else {
338        data = dataset.GetReadOnlyDoubleValues(variable.VariableName).ToArray();
339        cachedData[variable.VariableName] = (double[])instruction.data;
340      }
341
342      var paramIdx = -1;
343      if (node2paramIdx.ContainsKey(variable)) {
344        paramIdx = node2paramIdx[variable];
345        var f = new AlgebraicDoubleVector(BATCHSIZE);
346        var g = new AlgebraicDoubleVector(BATCHSIZE);
347        instruction.value = new MultivariateDual<AlgebraicDoubleVector>(f, paramIdx, g);
348      } else {
349        var f = new AlgebraicDoubleVector(BATCHSIZE);
350        instruction.value = new MultivariateDual<AlgebraicDoubleVector>(f);
351      }
352
353      instruction.dblVal = variable.Weight;
354      instruction.data = new object[] { data, paramIdx };
355    }
356
357    protected override void LoadVariable(Instruction a) {
358      var paramIdx = (int)((object[])a.data)[1];
359      var data = (double[])((object[])a.data)[0];
360
361      for (int i = rowIndex; i < rows.Length && i - rowIndex < BATCHSIZE; i++) a.value.Value[i - rowIndex] = data[rows[i]];
362      a.value.Scale(a.dblVal);
363
364      if (paramIdx >= 0) {
365        // update gradient with variable values
366        var g = a.value.Gradient.Elements[paramIdx];
367        for (int i = rowIndex; i < rows.Length && i - rowIndex < BATCHSIZE; i++) {
368          g[i - rowIndex] = data[rows[i]];
369        }
370      }
371    }
372  }
373
374
375  public sealed class IntervalEvaluator : Interpreter<AlgebraicInterval> {
376    [ThreadStatic]
377    private IDictionary<string, Interval> intervals;
378
379    public Interval Evaluate(ISymbolicExpressionTree tree, IDictionary<string, Interval> intervals) {
380      this.intervals = intervals;
381      var code = Compile(tree);
382      Evaluate(code);
383      return new Interval(code[0].value.LowerBound.Value.Value, code[0].value.UpperBound.Value.Value);
384    }
385
386    public Interval Evaluate(ISymbolicExpressionTree tree, IDictionary<string, Interval> intervals, ISymbolicExpressionTreeNode[] paramNodes, out double[] lowerGradient, out double[] upperGradient) {
387      this.intervals = intervals;
388      var code = Compile(tree);
389      Evaluate(code);
390      lowerGradient = new double[paramNodes.Length];
391      upperGradient = new double[paramNodes.Length];
392      var l = code[0].value.LowerBound;
393      var u = code[0].value.UpperBound;
394      for (int i = 0; i < paramNodes.Length; ++i) {
395        if (paramNodes[i] == null) continue;
396        if (l.Gradient.Elements.TryGetValue(paramNodes[i], out AlgebraicDouble value)) lowerGradient[i] = value;
397        if (u.Gradient.Elements.TryGetValue(paramNodes[i], out value)) upperGradient[i] = value;
398      }
399      return new Interval(code[0].value.LowerBound.Value.Value, code[0].value.UpperBound.Value.Value);
400    }
401
402    protected override void InitializeInternalInstruction(ref Instruction instruction, ISymbolicExpressionTreeNode node) {
403      instruction.value = new AlgebraicInterval(0, 0);
404    }
405
406
407    protected override void InitializeTerminalInstruction(ref Instruction instruction, ConstantTreeNode constant) {
408      instruction.dblVal = constant.Value;
409      instruction.value = new AlgebraicInterval(
410        new MultivariateDual<AlgebraicDouble>(constant.Value, constant, 1.0),
411        new MultivariateDual<AlgebraicDouble>(constant.Value, constant, 1.0) // use node as key
412        );
413    }
414
415    protected override void InitializeTerminalInstruction(ref Instruction instruction, VariableTreeNode variable) {
416      instruction.dblVal = variable.Weight;
417      instruction.value = new AlgebraicInterval(
418        low: new MultivariateDual<AlgebraicDouble>(intervals[variable.VariableName].LowerBound, variable, intervals[variable.VariableName].LowerBound),  // bounds change by variable value d/dc (c I(var)) = I(var)
419        high: new MultivariateDual<AlgebraicDouble>(intervals[variable.VariableName].UpperBound, variable, intervals[variable.VariableName].UpperBound)
420        );
421    }
422
423    protected override void LoadVariable(Instruction a) {
424      // nothing to do
425    }
426  }
427
428  public interface IAlgebraicType<T> {
429    T Zero { get; }
430    T One { get; }
431
432    T AssignAbs(T a); // set this to assign abs(a)
433    T Assign(T a); // assign this to same value as a (copy!)
434    T AssignNeg(T a); // set this to negative(a)
435    T AssignInv(T a); // set this to inv(a);
436    T Scale(double s); // scale this with s
437    T Add(T a); // add a to this
438    T Sub(T a); // subtract a from this
439    T Mul(T a); // multiply this with a
440    T Div(T a); // divide this by a
441    T AssignLog(T a); // set this to log a
442    T AssignExp(T a); // set this to exp(a)
443    T AssignSin(T a); // set this to sin(a)
444    T AssignCos(T a); // set this to cos(a)
445    T AssignTanh(T a); // set this to tanh(a)
446    T AssignIntPower(T a, int p);
447    T AssignIntRoot(T a, int r);
448    T AssignSgn(T a); // set this to sign(a)
449    T Clone();
450  }
451
452  public static class Algebraic {
453    public static T Abs<T>(this T a) where T : IAlgebraicType<T> { a.AssignAbs(a.Clone()); return a; }
454    public static T Neg<T>(this T a) where T : IAlgebraicType<T> { a.AssignNeg(a.Clone()); return a; }
455    public static T Inv<T>(this T a) where T : IAlgebraicType<T> { a.AssignInv(a.Clone()); return a; }
456    public static T Log<T>(this T a) where T : IAlgebraicType<T> { a.AssignLog(a.Clone()); return a; }
457    public static T Exp<T>(this T a) where T : IAlgebraicType<T> { a.AssignExp(a.Clone()); return a; }
458    public static T Sin<T>(this T a) where T : IAlgebraicType<T> { a.AssignSin(a.Clone()); return a; }
459    public static T Cos<T>(this T a) where T : IAlgebraicType<T> { a.AssignCos(a.Clone()); return a; }
460    public static T Sgn<T>(this T a) where T : IAlgebraicType<T> { a.AssignSgn(a.Clone()); return a; }
461    public static T IntPower<T>(this T a, int p) where T : IAlgebraicType<T> { a.AssignIntPower(a.Clone(), p); return a; }
462    public static T IntRoot<T>(this T a, int r) where T : IAlgebraicType<T> { a.AssignIntRoot(a.Clone(), r); return a; }
463
464    public static T Max<T>(T a, T b) where T : IAlgebraicType<T> {
465      // ((a + b) + abs(b - a)) / 2
466      return a.Clone().Add(b).Add(b.Clone().Sub(a).Abs()).Scale(1.0 / 2.0);
467    }
468    public static T Min<T>(T a, T b) where T : IAlgebraicType<T> {
469      // ((a + b) - abs(a - b)) / 2
470      return a.Clone().Add(b).Sub(a.Clone().Sub(b).Abs()).Scale(1.0 / 2.0);
471    }
472  }
473
474
475  // algebraic type wrapper for a double value
476  [DebuggerDisplay("{Value}")]
477  public sealed class AlgebraicDouble : IAlgebraicType<AlgebraicDouble> {
478    public static implicit operator AlgebraicDouble(double value) { return new AlgebraicDouble(value); }
479    public static implicit operator double(AlgebraicDouble value) { return value.Value; }
480    public double Value;
481
482    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
483    public AlgebraicDouble Zero => new AlgebraicDouble(0.0);
484    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
485    public AlgebraicDouble One => new AlgebraicDouble(1.0);
486    public AlgebraicDouble() { }
487    public AlgebraicDouble(double value) { this.Value = value; }
488    public AlgebraicDouble Assign(AlgebraicDouble a) { Value = a.Value; return this; }
489    public AlgebraicDouble Add(AlgebraicDouble a) { Value += a.Value; return this; }
490    public AlgebraicDouble Sub(AlgebraicDouble a) { Value -= a.Value; return this; }
491    public AlgebraicDouble Mul(AlgebraicDouble a) { Value *= a.Value; return this; }
492    public AlgebraicDouble Div(AlgebraicDouble a) { Value /= a.Value; return this; }
493    public AlgebraicDouble Scale(double s) { Value *= s; return this; }
494    public AlgebraicDouble AssignInv(AlgebraicDouble a) { Value = 1.0 / a.Value; return this; }
495    public AlgebraicDouble AssignNeg(AlgebraicDouble a) { Value = -a.Value; return this; }
496    public AlgebraicDouble AssignSin(AlgebraicDouble a) { Value = Math.Sin(a.Value); return this; }
497    public AlgebraicDouble AssignCos(AlgebraicDouble a) { Value = Math.Cos(a.Value); return this; }
498    public AlgebraicDouble AssignTanh(AlgebraicDouble a) { Value = Math.Tanh(a.Value); return this; }
499    public AlgebraicDouble AssignLog(AlgebraicDouble a) { Value = a.Value <= 0 ? double.NegativeInfinity : Math.Log(a.Value); return this; } // alternative definiton of log to prevent NaN
500    public AlgebraicDouble AssignExp(AlgebraicDouble a) { Value = Math.Exp(a.Value); return this; }
501    public AlgebraicDouble AssignIntPower(AlgebraicDouble a, int p) { Value = Math.Pow(a.Value, p); return this; }
502    public AlgebraicDouble AssignIntRoot(AlgebraicDouble a, int r) { Value = Math.Pow(a.Value, 1.0 / r); return this; }
503    public AlgebraicDouble AssignAbs(AlgebraicDouble a) { Value = Math.Abs(a.Value); return this; }
504    public AlgebraicDouble AssignSgn(AlgebraicDouble a) { Value = double.IsNaN(a.Value) ? double.NaN : Math.Sign(a.Value); return this; }
505    public AlgebraicDouble Clone() { return new AlgebraicDouble(Value); }
506
507    public override string ToString() {
508      return Value.ToString();
509    }
510  }
511
512  // a simple vector as an algebraic type
513  [DebuggerDisplay("DoubleVector(len={Length}): {string.}")]
514  public class AlgebraicDoubleVector : IAlgebraicType<AlgebraicDoubleVector> {
515    private double[] arr;
516    public double this[int idx] { get { return arr[idx]; } set { arr[idx] = value; } }
517    public int Length => arr.Length;
518
519    public AlgebraicDoubleVector(int length) { arr = new double[length]; }
520
521    public AlgebraicDoubleVector() { }
522
523    /// <summary>
524    ///
525    /// </summary>
526    /// <param name="arr">array is not copied</param>
527    public AlgebraicDoubleVector(double[] arr) { this.arr = arr; }
528
529    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
530    public AlgebraicDoubleVector Zero => new AlgebraicDoubleVector(new double[this.Length]); // must return vector of same length as this (therefore Zero is not static)
531    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
532    public AlgebraicDoubleVector One => new AlgebraicDoubleVector(new double[this.Length]).AssignConstant(1.0); // must return vector of same length as this (therefore Zero is not static)
533    public AlgebraicDoubleVector Assign(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = a.arr[i]; } return this; }
534    public AlgebraicDoubleVector Add(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] += a.arr[i]; } return this; }
535    public AlgebraicDoubleVector Sub(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] -= a.arr[i]; } return this; }
536    public AlgebraicDoubleVector Mul(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] *= a.arr[i]; } return this; }
537    public AlgebraicDoubleVector Div(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] /= a.arr[i]; } return this; }
538    public AlgebraicDoubleVector AssignNeg(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = -a.arr[i]; } return this; }
539    public AlgebraicDoubleVector AssignInv(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = 1.0 / a.arr[i]; } return this; }
540    public AlgebraicDoubleVector Scale(double s) { for (int i = 0; i < arr.Length; ++i) { arr[i] *= s; } return this; }
541    public AlgebraicDoubleVector AssignLog(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Log(a.arr[i]); } return this; }
542    public AlgebraicDoubleVector AssignSin(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Sin(a.arr[i]); } return this; }
543    public AlgebraicDoubleVector AssignExp(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Exp(a.arr[i]); } return this; }
544    public AlgebraicDoubleVector AssignCos(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Cos(a.arr[i]); } return this; }
545    public AlgebraicDoubleVector AssignTanh(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Tanh(a.arr[i]); } return this; }
546    public AlgebraicDoubleVector AssignIntPower(AlgebraicDoubleVector a, int p) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Pow(a.arr[i], p); } return this; }
547    public AlgebraicDoubleVector AssignIntRoot(AlgebraicDoubleVector a, int r) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Pow(a.arr[i], 1.0 / r); } return this; }
548    public AlgebraicDoubleVector AssignAbs(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Abs(a.arr[i]); } return this; }
549    public AlgebraicDoubleVector AssignSgn(AlgebraicDoubleVector a) { for (int i = 0; i < arr.Length; ++i) { arr[i] = Math.Sign(a.arr[i]); } return this; }
550
551    public AlgebraicDoubleVector Clone() {
552      var v = new AlgebraicDoubleVector(this.arr.Length);
553      Array.Copy(arr, v.arr, v.arr.Length);
554      return v;
555    }
556
557    public AlgebraicDoubleVector AssignConstant(double constantValue) {
558      for (int i = 0; i < arr.Length; ++i) {
559        arr[i] = constantValue;
560      }
561      return this;
562    }
563
564    public void CopyTo(double[] dest, int idx, int length) {
565      Array.Copy(arr, 0, dest, idx, length);
566    }
567
568    public void CopyFrom(double[] data, int rowIndex) {
569      Array.Copy(data, rowIndex, arr, 0, Math.Min(arr.Length, data.Length - rowIndex));
570    }
571    public void CopyRowTo(double[,] dest, int row) {
572      for (int j = 0; j < arr.Length; ++j) dest[row, j] = arr[j];
573    }
574
575    internal void CopyColumnTo(double[,] dest, int column, int row, int len) {
576      for (int j = 0; j < len; ++j) dest[row + j, column] = arr[j];
577    }
578
579    public override string ToString() {
580      return "{" + string.Join(", ", arr.Take(Math.Max(5, arr.Length))) + (arr.Length > 5 ? "..." : string.Empty) + "}";
581    }
582  }
583
584
585  /*
586  // vectors of algebraic types
587  public sealed class AlgebraicVector<T> : IAlgebraicType<AlgebraicVector<T>> where T : IAlgebraicType<T>, new() {
588    private T[] elems;
589
590    public T this[int idx] { get { return elems[idx]; } set { elems[idx] = value; } }
591
592    public int Length => elems.Length;
593
594    private AlgebraicVector() { }
595
596    public AlgebraicVector(int len) { elems = new T[len]; }
597
598    /// <summary>
599    ///
600    /// </summary>
601    /// <param name="elems">The array is copied (element-wise clone)</param>
602    public AlgebraicVector(T[] elems) {
603      this.elems = new T[elems.Length];
604      for (int i = 0; i < elems.Length; ++i) { this.elems[i] = elems[i].Clone(); }
605    }
606
607    /// <summary>
608    ///
609    /// </summary>
610    /// <param name="elems">Array is not copied!</param>
611    /// <returns></returns>
612    public AlgebraicVector<T> FromArray(T[] elems) {
613      var v = new AlgebraicVector<T>();
614      v.elems = elems;
615      return v;
616    }
617
618    public void CopyTo(T[] dest) {
619      if (dest.Length != elems.Length) throw new InvalidOperationException("arr lengths do not match in Vector<T>.Copy");
620      Array.Copy(elems, dest, dest.Length);
621    }
622
623    public AlgebraicVector<T> Clone() { return new AlgebraicVector<T>(elems); }
624
625
626    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
627    public AlgebraicVector<T> Zero => new AlgebraicVector<T>(Length);
628    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
629    public AlgebraicVector<T> One { get { var v = new AlgebraicVector<T>(Length); for (int i = 0; i < elems.Length; ++i) elems[i] = new T().One; return v; } }
630    public AlgebraicVector<T> Assign(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].Assign(a.elems[i]); } return this; }
631    public AlgebraicVector<T> Add(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].Add(a.elems[i]); } return this; }
632    public AlgebraicVector<T> Sub(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].Sub(a.elems[i]); } return this; }
633    public AlgebraicVector<T> Mul(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].Mul(a.elems[i]); } return this; }
634    public AlgebraicVector<T> Div(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].Div(a.elems[i]); } return this; }
635    public AlgebraicVector<T> AssignNeg(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignNeg(a.elems[i]); } return this; }
636    public AlgebraicVector<T> Scale(double s) { for (int i = 0; i < elems.Length; ++i) { elems[i].Scale(s); } return this; }
637    public AlgebraicVector<T> Scale(T s) { for (int i = 0; i < elems.Length; ++i) { elems[i].Mul(s); } return this; }
638    public AlgebraicVector<T> AssignInv(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignInv(a.elems[i]); } return this; }
639    public AlgebraicVector<T> AssignLog(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignLog(a.elems[i]); } return this; }
640    public AlgebraicVector<T> AssignExp(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignExp(a.elems[i]); } return this; }
641    public AlgebraicVector<T> AssignSin(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignSin(a.elems[i]); } return this; }
642    public AlgebraicVector<T> AssignCos(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignCos(a.elems[i]); } return this; }
643    public AlgebraicVector<T> AssignIntPower(AlgebraicVector<T> a, int p) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignIntPower(a.elems[i], p); } return this; }
644    public AlgebraicVector<T> AssignIntRoot(AlgebraicVector<T> a, int r) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignIntRoot(a.elems[i], r); } return this; }
645    public AlgebraicVector<T> AssignAbs(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignAbs(a.elems[i]); } return this; }
646    public AlgebraicVector<T> AssignSgn(AlgebraicVector<T> a) { for (int i = 0; i < elems.Length; ++i) { elems[i].AssignSgn(a.elems[i]); } return this; }
647  }
648
649  */
650
651
652  /// <summary>
653  /// A sparse vector of algebraic types. Elements are accessed via a key of type K
654  /// </summary>
655  /// <typeparam name="K">Key type</typeparam>
656  /// <typeparam name="T">Element type</typeparam>
657  [DebuggerDisplay("SparseVector: {ToString()}")]
658  public sealed class AlgebraicSparseVector<K, T> : IAlgebraicType<AlgebraicSparseVector<K, T>> where T : IAlgebraicType<T> {
659    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
660    private Dictionary<K, T> elems;
661    public IReadOnlyDictionary<K, T> Elements => elems;
662
663
664    public AlgebraicSparseVector(AlgebraicSparseVector<K, T> original) {
665      elems = original.elems.ToDictionary(kvp => kvp.Key, kvp => kvp.Value.Clone());
666    }
667
668    /// <summary>
669    ///
670    /// </summary>
671    /// <param name="keys"></param>
672    /// <param name="values">values are cloned</param>
673    public AlgebraicSparseVector(K[] keys, T[] values) {
674      if (keys.Length != values.Length) throw new ArgumentException("lengths of keys and values doesn't match in SparseVector");
675      elems = new Dictionary<K, T>(keys.Length);
676      for (int i = 0; i < keys.Length; ++i) {
677        elems.Add(keys[i], values[i].Clone());
678      }
679    }
680
681    public AlgebraicSparseVector() {
682      this.elems = new Dictionary<K, T>();
683    }
684
685    // keep only elements from a
686    private void AssignFromSource(AlgebraicSparseVector<K, T> a, Func<T, T, T> mapAssign) {
687      // remove elems from this which don't occur in a
688      List<K> keysToRemove = new List<K>();
689      foreach (var kvp in elems) {
690        if (!a.elems.ContainsKey(kvp.Key)) keysToRemove.Add(kvp.Key);
691      }
692      foreach (var o in keysToRemove) elems.Remove(o); // -> zero
693
694      foreach (var kvp in a.elems) {
695        if (elems.TryGetValue(kvp.Key, out T value))
696          mapAssign(kvp.Value, value);
697        else
698          elems.Add(kvp.Key, mapAssign(kvp.Value, kvp.Value.Zero));
699      }
700    }
701
702    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
703    public AlgebraicSparseVector<K, T> Zero => new AlgebraicSparseVector<K, T>();
704    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
705    public AlgebraicSparseVector<K, T> One => throw new NotSupportedException();
706
707    public AlgebraicSparseVector<K, T> Scale(T s) { foreach (var kvp in elems) { kvp.Value.Mul(s); } return this; }
708    public AlgebraicSparseVector<K, T> Scale(double s) { foreach (var kvp in elems) { kvp.Value.Scale(s); } return this; }
709
710    public AlgebraicSparseVector<K, T> Assign(AlgebraicSparseVector<K, T> a) { elems.Clear(); AssignFromSource(a, (src, dest) => dest.Assign(src)); return this; }
711    public AlgebraicSparseVector<K, T> AssignInv(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignInv(src)); return this; }
712    public AlgebraicSparseVector<K, T> AssignNeg(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignNeg(src)); return this; }
713    public AlgebraicSparseVector<K, T> AssignLog(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignLog(src)); return this; }
714    public AlgebraicSparseVector<K, T> AssignExp(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignExp(src)); return this; }
715    public AlgebraicSparseVector<K, T> AssignIntPower(AlgebraicSparseVector<K, T> a, int p) { AssignFromSource(a, (src, dest) => dest.AssignIntPower(src, p)); return this; }
716    public AlgebraicSparseVector<K, T> AssignIntRoot(AlgebraicSparseVector<K, T> a, int r) { AssignFromSource(a, (src, dest) => dest.AssignIntRoot(src, r)); return this; }
717    public AlgebraicSparseVector<K, T> AssignSin(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignSin(src)); return this; }
718    public AlgebraicSparseVector<K, T> AssignCos(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignCos(src)); return this; }
719    public AlgebraicSparseVector<K, T> AssignTanh(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignTanh(src)); return this; }
720    public AlgebraicSparseVector<K, T> AssignAbs(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignAbs(src)); return this; }
721    public AlgebraicSparseVector<K, T> AssignSgn(AlgebraicSparseVector<K, T> a) { AssignFromSource(a, (src, dest) => dest.AssignSgn(src)); return this; }
722    public AlgebraicSparseVector<K, T> Add(AlgebraicSparseVector<K, T> a) {
723      foreach (var kvp in a.elems) {
724        if (elems.TryGetValue(kvp.Key, out T value))
725          value.Add(kvp.Value);
726        else
727          elems.Add(kvp.Key, kvp.Value.Clone()); // 0 + a
728      }
729      return this;
730    }
731
732    public AlgebraicSparseVector<K, T> Sub(AlgebraicSparseVector<K, T> a) {
733      foreach (var kvp in a.elems) {
734        if (elems.TryGetValue(kvp.Key, out T value))
735          value.Sub(kvp.Value);
736        else
737          elems.Add(kvp.Key, kvp.Value.Zero.Sub(kvp.Value)); // 0 - a
738      }
739      return this;
740    }
741
742    public AlgebraicSparseVector<K, T> Mul(AlgebraicSparseVector<K, T> a) {
743      var keys = elems.Keys.ToArray();
744      foreach (var k in keys) if (!a.elems.ContainsKey(k)) elems.Remove(k); // 0 * a => 0
745      foreach (var kvp in a.elems) {
746        if (elems.TryGetValue(kvp.Key, out T value))
747          value.Mul(kvp.Value); // this * a
748      }
749      return this;
750    }
751
752    public AlgebraicSparseVector<K, T> Div(AlgebraicSparseVector<K, T> a) {
753      return Mul(a.Clone().Inv());
754    }
755
756    public AlgebraicSparseVector<K, T> Clone() {
757      return new AlgebraicSparseVector<K, T>(this);
758    }
759
760    public override string ToString() {
761      return "[" + string.Join(" ", elems.Select(kvp => kvp.Key + ": " + kvp.Value)) + "]";
762    }
763  }
764
765  [DebuggerDisplay("[{low.Value}..{high.Value}]")]
766  public class AlgebraicInterval : IAlgebraicType<AlgebraicInterval> {
767    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
768    private MultivariateDual<AlgebraicDouble> low;
769    public MultivariateDual<AlgebraicDouble> LowerBound => low.Clone();
770
771    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
772    private MultivariateDual<AlgebraicDouble> high;
773    public MultivariateDual<AlgebraicDouble> UpperBound => high.Clone();
774
775
776    public AlgebraicInterval() : this(double.NegativeInfinity, double.PositiveInfinity) { }
777
778    public AlgebraicInterval(MultivariateDual<AlgebraicDouble> low, MultivariateDual<AlgebraicDouble> high) {
779      this.low = low.Clone();
780      this.high = high.Clone();
781    }
782
783    public AlgebraicInterval(double low, double high) {
784      this.low = new MultivariateDual<AlgebraicDouble>(new AlgebraicDouble(low));
785      this.high = new MultivariateDual<AlgebraicDouble>(new AlgebraicDouble(high));
786    }
787
788    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
789    public AlgebraicInterval Zero => new AlgebraicInterval(0.0, 0.0);
790    public AlgebraicInterval One => new AlgebraicInterval(1.0, 1.0);
791    public AlgebraicInterval Add(AlgebraicInterval a) {
792      low.Add(a.low);
793      high.Add(a.high);
794      return this;
795    }
796
797    public AlgebraicInterval Mul(AlgebraicInterval a) {
798      var v1 = low.Clone().Mul(a.low);
799      var v2 = low.Clone().Mul(a.high);
800      var v3 = high.Clone().Mul(a.low);
801      var v4 = high.Clone().Mul(a.high);
802
803      low = Algebraic.Min(Algebraic.Min(v1, v2), Algebraic.Min(v3, v4));
804      high = Algebraic.Max(Algebraic.Max(v1, v2), Algebraic.Max(v3, v4));
805      return this;
806    }
807
808    public AlgebraicInterval Assign(AlgebraicInterval a) {
809      low = a.low;
810      high = a.high;
811      return this;
812    }
813
814    public AlgebraicInterval AssignCos(AlgebraicInterval a) {
815      return AssignSin(a.Clone().Sub(new AlgebraicInterval(Math.PI / 2, Math.PI / 2)));
816    }
817
818    public AlgebraicInterval Div(AlgebraicInterval a) {
819      if (a.Contains(0.0)) {
820        if (a.low.Value.Value.IsAlmost(0.0) && a.high.Value.Value.IsAlmost(0.0)) {
821          low = new MultivariateDual<AlgebraicDouble>(double.NegativeInfinity);
822          high = new MultivariateDual<AlgebraicDouble>(double.PositiveInfinity);
823        } else if (a.low.Value.Value.IsAlmost(0.0))
824          Mul(new AlgebraicInterval(a.Clone().high.Inv(), new MultivariateDual<AlgebraicDouble>(double.PositiveInfinity)));
825        else
826          Mul(new AlgebraicInterval(new MultivariateDual<AlgebraicDouble>(double.NegativeInfinity), a.low.Clone().Inv()));
827      } else {
828        Mul(new AlgebraicInterval(a.high.Clone().Inv(), a.low.Clone().Inv())); // inverting leads to inverse roles of high and low
829      }
830      return this;
831    }
832
833    public AlgebraicInterval AssignExp(AlgebraicInterval a) {
834      low.AssignExp(a.low);
835      high.AssignExp(a.high);
836      return this;
837    }
838
839    // tanh is a bijective function
840    public AlgebraicInterval AssignTanh(AlgebraicInterval a) {
841      low.AssignTanh(a.low);
842      high.AssignTanh(a.high);
843      return this;
844    }
845
846    public AlgebraicInterval AssignIntPower(AlgebraicInterval a, int p) {
847      if (p < 0) {  // x^-3 == 1/(x^3)
848        AssignIntPower(a, -p);
849        return AssignInv(this);
850      } else if (p == 0) {
851        if (a.Contains(0.0)) {
852          // => 0^0 = 0 ; might not be relevant
853          low = new MultivariateDual<AlgebraicDouble>(0.0);
854          high = new MultivariateDual<AlgebraicDouble>(1.0);
855          return this;
856        } else {
857          // => 1
858          low = new MultivariateDual<AlgebraicDouble>(1.0);
859          high = new MultivariateDual<AlgebraicDouble>(1.0);
860          return this;
861        }
862      } else if (p == 1) return this;
863      else if (p % 2 == 0) {
864        // p is even => interval must be positive
865        if (a.Contains(0.0)) {
866          low = new MultivariateDual<AlgebraicDouble>(0.0);
867          high = Algebraic.Max(a.low.IntPower(p), a.high.IntPower(p));
868        } else {
869          var lowPower = a.low.IntPower(p);
870          var highPower = a.high.IntPower(p);
871          low = Algebraic.Min(lowPower, highPower);
872          high = Algebraic.Max(lowPower, highPower);
873        }
874      } else {
875        // p is uneven
876        if (a.Contains(0.0)) {
877          low.AssignIntPower(a.low, p);
878          high.AssignIntPower(a.high, p);
879        } else {
880          var lowPower = a.low.IntPower(p);
881          var highPower = a.high.IntPower(p);
882          low = Algebraic.Min(lowPower, highPower);
883          high = Algebraic.Max(lowPower, highPower);
884        }
885      }
886      return this;
887    }
888
889    public AlgebraicInterval AssignIntRoot(AlgebraicInterval a, int r) {
890      if (r == 0) { low = new MultivariateDual<AlgebraicDouble>(double.NaN); high = new MultivariateDual<AlgebraicDouble>(double.NaN); return this; }
891      if (r == 1) return this;
892      if (r < 0) {
893        // x^ (-1/2) = 1 / (x^(1/2))
894        AssignIntRoot(a, -r);
895        return AssignInv(this);
896      } else {
897        // root only defined for positive arguments
898        if (a.LowerBound.Value.Value < 0) {
899          low = new MultivariateDual<AlgebraicDouble>(double.NaN);
900          high = new MultivariateDual<AlgebraicDouble>(double.NaN);
901          return this;
902        } else {
903          low.AssignIntRoot(a.low, r);
904          high.AssignIntRoot(a.high, r);
905          return this;
906        }
907      }
908    }
909
910    public AlgebraicInterval AssignInv(AlgebraicInterval a) {
911      low = new MultivariateDual<AlgebraicDouble>(1.0);
912      high = new MultivariateDual<AlgebraicDouble>(1.0);
913      return Div(a);
914    }
915
916    public AlgebraicInterval AssignLog(AlgebraicInterval a) {
917      low.AssignLog(a.low);
918      high.AssignLog(a.high);
919      return this;
920    }
921
922    public AlgebraicInterval AssignNeg(AlgebraicInterval a) {
923      low.AssignNeg(a.high);
924      high.AssignNeg(a.low);
925      return this;
926    }
927
928    public AlgebraicInterval Scale(double s) {
929      low.Scale(s);
930      high.Scale(s);
931      if (s < 0) {
932        var t = low;
933        low = high;
934        high = t;
935      }
936      return this;
937    }
938
939    public AlgebraicInterval AssignSin(AlgebraicInterval a) {
940      if (Math.Abs(a.UpperBound.Value.Value - a.LowerBound.Value.Value) >= Math.PI * 2) {
941        low = new MultivariateDual<AlgebraicDouble>(-1.0);
942        high = new MultivariateDual<AlgebraicDouble>(1.0);
943      }
944
945      //divide the interval by PI/2 so that the optima lie at x element of N (0,1,2,3,4,...)
946      double Pihalf = Math.PI / 2;
947      var scaled = this.Clone().Scale(1.0 / Pihalf);
948      //move to positive scale
949      if (scaled.LowerBound.Value.Value < 0) {
950        int periodsToMove = Math.Abs((int)scaled.LowerBound.Value.Value / 4) + 1;
951        scaled.Add(new AlgebraicInterval(periodsToMove * 4, periodsToMove * 4));
952      }
953
954      double scaledLowerBound = scaled.LowerBound.Value.Value % 4.0;
955      double scaledUpperBound = scaled.UpperBound.Value.Value % 4.0;
956      if (scaledUpperBound < scaledLowerBound) scaledUpperBound += 4.0;
957      List<double> sinValues = new List<double>();
958      sinValues.Add(Math.Sin(scaledLowerBound * Pihalf));
959      sinValues.Add(Math.Sin(scaledUpperBound * Pihalf));
960
961      int startValue = (int)Math.Ceiling(scaledLowerBound);
962      while (startValue < scaledUpperBound) {
963        sinValues.Add(Math.Sin(startValue * Pihalf));
964        startValue += 1;
965      }
966
967      low = new MultivariateDual<AlgebraicDouble>(sinValues.Min());
968      high = new MultivariateDual<AlgebraicDouble>(sinValues.Max());
969      return this;
970    }
971
972    public AlgebraicInterval Sub(AlgebraicInterval a) {
973      // [x1,x2] − [y1,y2] = [x1 − y2,x2 − y1]
974      low.Sub(a.high);
975      high.Sub(a.low);
976      return this;
977    }
978
979    public AlgebraicInterval Clone() {
980      return new AlgebraicInterval(low, high);
981    }
982
983    public bool Contains(double val) {
984      return LowerBound.Value.Value <= val && val <= UpperBound.Value.Value;
985    }
986
987    public AlgebraicInterval AssignAbs(AlgebraicInterval a) {
988      if (a.Contains(0.0)) {
989        var abslow = a.low.Clone().Abs();
990        var abshigh = a.high.Clone().Abs();
991        a.high.Assign(Algebraic.Max(abslow, abshigh));
992        a.low.Assign(new MultivariateDual<AlgebraicDouble>(0.0)); // lost gradient for lower bound
993      } else {
994        var abslow = a.low.Clone().Abs();
995        var abshigh = a.high.Clone().Abs();
996        a.low.Assign(Algebraic.Min(abslow, abshigh));
997        a.high.Assign(Algebraic.Max(abslow, abshigh));
998      }
999      return this;
1000    }
1001
1002    public AlgebraicInterval AssignSgn(AlgebraicInterval a) {
1003      low.AssignSgn(a.low);
1004      high.AssignSgn(a.high);
1005      return this;
1006    }
1007  }
1008
1009  public class Dual<V> : IAlgebraicType<Dual<V>>
1010    where V : IAlgebraicType<V> {
1011    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1012    private V v;
1013    public V Value => v;
1014
1015    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1016    private V dv;
1017    public V Derivative => dv;
1018
1019    public Dual(V v, V dv) { this.v = v; this.dv = dv; }
1020
1021    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1022    public Dual<V> Zero => new Dual<V>(Value.Zero, Derivative.Zero);
1023    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1024    public Dual<V> One => new Dual<V>(Value.One, Derivative.Zero);
1025
1026    public Dual<V> Assign(Dual<V> a) { v.Assign(a.v); dv.Assign(a.dv); return this; }
1027    public Dual<V> Scale(double s) { v.Scale(s); dv.Scale(s); return this; }
1028    public Dual<V> Add(Dual<V> a) { v.Add(a.v); dv.Add(a.dv); return this; }
1029    public Dual<V> Sub(Dual<V> a) { v.Sub(a.v); dv.Sub(a.dv); return this; }
1030    public Dual<V> AssignNeg(Dual<V> a) { v.AssignNeg(a.v); dv.AssignNeg(a.dv); return this; }
1031    public Dual<V> AssignInv(Dual<V> a) { v.AssignInv(a.v); dv.AssignNeg(a.dv).Mul(v).Mul(v); return this; } // (1/f(x))' = - f(x)' / f(x)^2
1032
1033    // (a(x) * b(x))' = b(x)*a(x)' + b(x)'*a(x);
1034    public Dual<V> Mul(Dual<V> a) {
1035      var t1 = a.dv.Clone().Mul(v);
1036      var t2 = dv.Clone().Mul(a.v);
1037      dv.Assign(t1).Add(t2);
1038
1039      v.Mul(a.v);
1040      return this;
1041    }
1042    public Dual<V> Div(Dual<V> a) { Mul(a.Inv()); return this; }
1043
1044    public Dual<V> AssignExp(Dual<V> a) { v.AssignExp(a.v); dv.Assign(a.dv).Mul(v); return this; } // exp(f(x)) = exp(f(x))*f(x)'
1045    public Dual<V> AssignLog(Dual<V> a) { v.AssignLog(a.v); dv.Assign(a.dv).Div(a.v); return this; }     // log(x)' = 1/f(x) * f(x)'
1046
1047    public Dual<V> AssignIntPower(Dual<V> a, int p) { v.AssignIntPower(a.v, p); dv.Assign(a.dv).Scale(p).Mul(a.v.Clone().IntPower(p - 1)); return this; }
1048    public Dual<V> AssignIntRoot(Dual<V> a, int r) { v.AssignIntRoot(a.v, r); dv.Assign(a.dv).Scale(1.0 / r).Mul(a.v.IntRoot(r - 1)); return this; }
1049
1050    public Dual<V> AssignSin(Dual<V> a) { v.AssignSin(a.v); dv.Assign(a.dv).Mul(a.v.Clone().Cos()); return this; }
1051    public Dual<V> AssignCos(Dual<V> a) { v.AssignCos(a.v); dv.AssignNeg(a.dv).Mul(a.v.Clone().Sin()); return this; }
1052    public Dual<V> AssignTanh(Dual<V> a) { v.AssignTanh(a.v); dv.Assign(a.dv.Mul(v.Clone().IntPower(2).Neg().Add(Value.One))); return this; }
1053
1054    public Dual<V> AssignAbs(Dual<V> a) { v.AssignAbs(a.v); dv.Assign(a.dv).Mul(a.v.Clone().Sgn()); return this; }       // abs(f(x))' = f(x)*f'(x) / |f(x)|     
1055    public Dual<V> AssignSgn(Dual<V> a) { v.AssignSgn(a.v); dv.Assign(a.dv.Zero); return this; }
1056
1057    public Dual<V> Clone() { return new Dual<V>(v.Clone(), dv.Clone()); }
1058
1059  }
1060
1061  /// <summary>
1062  /// An algebraic type which has a value as well as the partial derivatives of the value over multiple variables.
1063  /// </summary>
1064  /// <typeparam name="V"></typeparam>
1065  [DebuggerDisplay("v={Value}; dv={dv}")]
1066  public class MultivariateDual<V> : IAlgebraicType<MultivariateDual<V>> where V : IAlgebraicType<V>, new() {
1067    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1068    private V v;
1069    public V Value => v;
1070
1071    [DebuggerBrowsable(DebuggerBrowsableState.Never)]
1072    private AlgebraicSparseVector<object, V> dv;
1073    public AlgebraicSparseVector<object, V> Gradient => dv; // <key,value> partial derivative identified via the key
1074
1075    private MultivariateDual(MultivariateDual<V> orig) { this.v = orig.v.Clone(); this.dv = orig.dv.Clone(); }
1076
1077    /// <summary>
1078    /// Constructor without partial derivative
1079    /// </summary>
1080    /// <param name="v"></param>
1081    public MultivariateDual(V v) { this.v = v.Clone(); this.dv = new AlgebraicSparseVector<object, V>(); }
1082
1083    /// <summary>
1084    /// Constructor for multiple partial derivatives
1085    /// </summary>
1086    /// <param name="v"></param>
1087    /// <param name="keys"></param>
1088    /// <param name="dv"></param>
1089    public MultivariateDual(V v, object[] keys, V[] dv) { this.v = v.Clone(); this.dv = new AlgebraicSparseVector<object, V>(keys, dv); }
1090
1091    /// <summary>
1092    /// Constructor for a single partial derivative
1093    /// </summary>
1094    /// <param name="v"></param>
1095    /// <param name="key"></param>
1096    /// <param name="dv"></param>
1097    public MultivariateDual(V v, object key, V dv) { this.v = v.Clone(); this.dv = new AlgebraicSparseVector<object, V>(new[] { key }, new[] { dv }); }
1098
1099    /// <summary>
1100    /// Constructor with a given value and gradient. For internal use.
1101    /// </summary>
1102    /// <param name="v">The value (not cloned).</param>
1103    /// <param name="gradient">The gradient (not cloned).</param>
1104    internal MultivariateDual(V v, AlgebraicSparseVector<object, V> gradient) { this.v = v; this.dv = gradient; }
1105
1106    public MultivariateDual<V> Clone() { return new MultivariateDual<V>(this); }
1107
1108    public MultivariateDual<V> Zero => new MultivariateDual<V>(Value.Zero, Gradient.Zero);
1109    public MultivariateDual<V> One => new MultivariateDual<V>(Value.One, Gradient.Zero);
1110
1111    public MultivariateDual<V> Scale(double s) { v.Scale(s); dv.Scale(s); return this; }
1112
1113    public MultivariateDual<V> Add(MultivariateDual<V> a) { v.Add(a.v); dv.Add(a.dv); return this; }
1114    public MultivariateDual<V> Sub(MultivariateDual<V> a) { v.Sub(a.v); dv.Sub(a.dv); return this; }
1115    public MultivariateDual<V> Assign(MultivariateDual<V> a) { v.Assign(a.v); dv.Assign(a.dv); return this; }
1116    public MultivariateDual<V> Mul(MultivariateDual<V> a) {
1117      // (a(x) * b(x))' = b(x)*a(x)' + b(x)'*a(x);
1118      var t1 = a.dv.Clone().Scale(v);
1119      var t2 = dv.Clone().Scale(a.v);
1120      dv.Assign(t1).Add(t2);
1121
1122      v.Mul(a.v);
1123      return this;
1124    }
1125    public MultivariateDual<V> Div(MultivariateDual<V> a) { v.Div(a.v); dv.Mul(a.dv.Inv()); return this; }
1126    public MultivariateDual<V> AssignNeg(MultivariateDual<V> a) { v.AssignNeg(a.v); dv.AssignNeg(a.dv); return this; }
1127    public MultivariateDual<V> AssignInv(MultivariateDual<V> a) { v.AssignInv(a.v); dv.AssignNeg(a.dv).Scale(v).Scale(v); return this; }   // (1/f(x))' = - f(x)' / f(x)^2
1128
1129    public MultivariateDual<V> AssignSin(MultivariateDual<V> a) { v.AssignSin(a.v); dv.Assign(a.dv).Scale(a.v.Clone().Cos()); return this; }
1130    public MultivariateDual<V> AssignCos(MultivariateDual<V> a) { v.AssignCos(a.v); dv.AssignNeg(a.dv).Scale(a.v.Clone().Sin()); return this; }
1131    public MultivariateDual<V> AssignTanh(MultivariateDual<V> a) { v.AssignTanh(a.v); dv.Assign(a.dv.Scale(v.Clone().IntPower(2).Neg().Add(Value.One))); return this; }     // tanh(f(x))' = f(x)'sech²(f(x)) = f(x)'(1 - tanh²(f(x)))
1132
1133    public MultivariateDual<V> AssignIntPower(MultivariateDual<V> a, int p) { v.AssignIntPower(a.v, p); dv.Assign(a.dv).Scale(p).Scale(a.v.Clone().IntPower(p - 1)); return this; }
1134    public MultivariateDual<V> AssignIntRoot(MultivariateDual<V> a, int r) { v.AssignIntRoot(a.v, r); dv.Assign(a.dv).Scale(1.0 / r).Scale(a.v.IntRoot(r - 1)); return this; }
1135
1136    public MultivariateDual<V> AssignExp(MultivariateDual<V> a) { v.AssignExp(a.v); dv.Assign(a.dv).Scale(v); return this; } // exp(f(x)) = exp(f(x))*f(x)'     
1137    public MultivariateDual<V> AssignLog(MultivariateDual<V> a) { v.AssignLog(a.v); dv.Assign(a.dv).Scale(a.v.Clone().Inv()); return this; }     // log(x)' = 1/f(x) * f(x)'
1138
1139    public MultivariateDual<V> AssignAbs(MultivariateDual<V> a) { v.AssignAbs(a.v); dv.Assign(a.dv).Scale(a.v.Clone().Sgn()); return this; }      // abs(f(x))' = f(x)*f'(x) / |f(x)|  doesn't work for intervals     
1140    public MultivariateDual<V> AssignSgn(MultivariateDual<V> a) { v.AssignSgn(a.v); dv = a.dv.Zero; return this; } // sign(f(x))' = 0;     
1141  }
1142}
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