mol-math: added EVD code

src/mol-math/linear-algebra/matrix/evd.ts

+/**
+ * Copyright (c) 2019 mol* contributors, licensed under MIT, See LICENSE file for more info.
+ *
+ * @author David Sehnal <david.sehnal@gmail.com>
+ */
+
+import Matrix from './matrix';
+
+export namespace EVD {
+    export interface Cache {
+        size: number,
+        matrix: Matrix,
+        eigenValues: number[],
+        D: number[],
+        E: number[]
+    }
+
+    export function createCache(size: number): Cache {
+        return {
+            size,
+            matrix: Matrix.create(size, size),
+            eigenValues: <any>new Float64Array(size),
+            D: <any>new Float64Array(size),
+            E: <any>new Float64Array(size)
+        }
+    }
+
+    /**
+     * Computes EVD and stores the result in the cache.
+     */
+    export function compute(cache: Cache): void {
+        symmetricEigenDecomp(cache.size, cache.matrix.data as number[], cache.eigenValues, cache.D, cache.E);
+    }
+}
+
+// The EVD code has been adapted from Math.NET, MIT license, Copyright (c) 2002-2015 Math.NET
+//
+// THE SOFTWARE IS PROVIDED 'AS IS', WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
+// INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR
+// PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE
+// FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE,
+// ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+
+function symmetricEigenDecomp(order: number, matrixEv: number[], vectorEv: number[], d: number[], e: number[]) {
+    for (let i = 0; i < order; i++) {
+        e[i] = 0.0;
+    }
+
+    let om1 = order - 1;
+    for (let i = 0; i < order; i++) {
+        d[i] = matrixEv[i * order + om1];
+    }
+
+    symmetricTridiagonalize(matrixEv, d, e, order);
+    symmetricDiagonalize(matrixEv, d, e, order);
+
+    for (let i = 0; i < order; i++) {
+        vectorEv[i] = d[i];
+    }
+}
+
+function symmetricTridiagonalize(a: number[], d: number[], e: number[], order: number) {
+    // Householder reduction to tridiagonal form.
+    for (let i = order - 1; i > 0; i--) {
+        // Scale to avoid under/overflow.
+        let scale = 0.0;
+        let h = 0.0;
+
+        for (let k = 0; k < i; k++) {
+            scale = scale + Math.abs(d[k]);
+        }
+
+        if (scale === 0.0) {
+            e[i] = d[i - 1];
+            for (let j = 0; j < i; j++) {
+                d[j] = a[(j * order) + i - 1];
+                a[(j * order) + i] = 0.0;
+                a[(i * order) + j] = 0.0;
+            }
+        }
+        else {
+            // Generate Householder vector.
+            for (let k = 0; k < i; k++) {
+                d[k] /= scale;
+                h += d[k] * d[k];
+            }
+
+            let f = d[i - 1];
+            let g = Math.sqrt(h);
+            if (f > 0) {
+                g = -g;
+            }
+
+            e[i] = scale * g;
+            h = h - (f * g);
+            d[i - 1] = f - g;
+
+            for (let j = 0; j < i; j++) {
+                e[j] = 0.0;
+            }
+
+            // Apply similarity transformation to remaining columns.
+            for (let j = 0; j < i; j++) {
+                f = d[j];
+                a[(i * order) + j] = f;
+                g = e[j] + (a[(j * order) + j] * f);
+
+                for (let k = j + 1; k <= i - 1; k++) {
+                    g += a[(j * order) + k] * d[k];
+                    e[k] += a[(j * order) + k] * f;
+                }
+
+                e[j] = g;
+            }
+
+            f = 0.0;
+
+            for (let j = 0; j < i; j++) {
+                e[j] /= h;
+                f += e[j] * d[j];
+            }
+
+            let hh = f / (h + h);
+
+            for (let j = 0; j < i; j++) {
+                e[j] -= hh * d[j];
+            }
+
+            for (let j = 0; j < i; j++) {
+                f = d[j];
+                g = e[j];
+
+                for (let k = j; k <= i - 1; k++) {
+                    a[(j * order) + k] -= (f * e[k]) + (g * d[k]);
+                }
+
+                d[j] = a[(j * order) + i - 1];
+                a[(j * order) + i] = 0.0;
+            }
+        }
+
+        d[i] = h;
+    }
+
+    // Accumulate transformations.
+    for (let i = 0; i < order - 1; i++) {
+        a[(i * order) + order - 1] = a[(i * order) + i];
+        a[(i * order) + i] = 1.0;
+        let h = d[i + 1];
+        if (h !== 0.0) {
+            for (let k = 0; k <= i; k++) {
+                d[k] = a[((i + 1) * order) + k] / h;
+            }
+
+            for (let j = 0; j <= i; j++) {
+                let g = 0.0;
+                for (let k = 0; k <= i; k++) {
+                    g += a[((i + 1) * order) + k] * a[(j * order) + k];
+                }
+
+                for (let k = 0; k <= i; k++) {
+                    a[(j * order) + k] -= g * d[k];
+                }
+            }
+        }
+
+        for (let k = 0; k <= i; k++) {
+            a[((i + 1) * order) + k] = 0.0;
+        }
+    }
+
+    for (let j = 0; j < order; j++) {
+        d[j] = a[(j * order) + order - 1];
+        a[(j * order) + order - 1] = 0.0;
+    }
+
+    a[(order * order) - 1] = 1.0;
+    e[0] = 0.0;
+}
+
+function symmetricDiagonalize(a: number[], d: number[], e: number[], order: number) {
+    const maxiter = 1000;
+
+    for (let i = 1; i < order; i++) {
+        e[i - 1] = e[i];
+    }
+
+    e[order - 1] = 0.0;
+
+    let f = 0.0;
+    let tst1 = 0.0;
+    let eps = Math.pow(2, -53); // DoubleWidth = 53
+    for (let l = 0; l < order; l++) {
+        // Find small subdiagonal element
+        tst1 = Math.max(tst1, Math.abs(d[l]) + Math.abs(e[l]));
+        let m = l;
+        while (m < order) {
+            if (Math.abs(e[m]) <= eps * tst1) {
+                break;
+            }
+
+            m++;
+        }
+
+        // If m == l, d[l] is an eigenvalue,
+        // otherwise, iterate.
+        if (m > l) {
+            let iter = 0;
+            do {
+                iter = iter + 1; // (Could check iteration count here.)
+
+                // Compute implicit shift
+                let g = d[l];
+                let p = (d[l + 1] - g) / (2.0 * e[l]);
+                let r = hypotenuse(p, 1.0);
+                if (p < 0) {
+                    r = -r;
+                }
+
+                d[l] = e[l] / (p + r);
+                d[l + 1] = e[l] * (p + r);
+
+                let dl1 = d[l + 1];
+                let h = g - d[l];
+                for (let i = l + 2; i < order; i++) {
+                    d[i] -= h;
+                }
+
+                f = f + h;
+
+                // Implicit QL transformation.
+                p = d[m];
+                let c = 1.0;
+                let c2 = c;
+                let c3 = c;
+                let el1 = e[l + 1];
+                let s = 0.0;
+                let s2 = 0.0;
+                for (let i = m - 1; i >= l; i--) {
+                    c3 = c2;
+                    c2 = c;
+                    s2 = s;
+                    g = c * e[i];
+                    h = c * p;
+                    r = hypotenuse(p, e[i]);
+                    e[i + 1] = s * r;
+                    s = e[i] / r;
+                    c = p / r;
+                    p = (c * d[i]) - (s * g);
+                    d[i + 1] = h + (s * ((c * g) + (s * d[i])));
+
+                    // Accumulate transformation.
+                    for (let k = 0; k < order; k++) {
+                        h = a[((i + 1) * order) + k];
+                        a[((i + 1) * order) + k] = (s * a[(i * order) + k]) + (c * h);
+                        a[(i * order) + k] = (c * a[(i * order) + k]) - (s * h);
+                    }
+                }
+
+                p = (-s) * s2 * c3 * el1 * e[l] / dl1;
+                e[l] = s * p;
+                d[l] = c * p;
+
+                // Check for convergence. If too many iterations have been performed,
+                // throw exception that Convergence Failed
+                if (iter >= maxiter) {
+                    throw 'SVD: Not converging.';
+                }
+            } while (Math.abs(e[l]) > eps * tst1);
+        }
+
+        d[l] = d[l] + f;
+        e[l] = 0.0;
+    }
+
+    // Sort eigenvalues and corresponding vectors.
+    for (let i = 0; i < order - 1; i++) {
+        let k = i;
+        let p = d[i];
+        for (let j = i + 1; j < order; j++) {
+            if (d[j] < p) {
+                k = j;
+                p = d[j];
+            }
+        }
+
+        if (k !== i) {
+            d[k] = d[i];
+            d[i] = p;
+            for (let j = 0; j < order; j++) {
+                p = a[(i * order) + j];
+                a[(i * order) + j] = a[(k * order) + j];
+                a[(k * order) + j] = p;
+            }
+        }
+    }
+}
+
+function hypotenuse(a: number, b: number) {
+    if (Math.abs(a) > Math.abs(b)) {
+        let r = b / a;
+        return Math.abs(a) * Math.sqrt(1 + (r * r));
+    }
+
+    if (b !== 0.0) {
+        let r = a / b;
+        return Math.abs(b) * Math.sqrt(1 + (r * r));
+    }
+
+    return 0.0;
+}
\ No newline at end of file

src/mol-math/linear-algebra/matrix/matrix.ts

@@ -14,6 +14,17 @@ namespace Matrix {
         return { data: new ctor(size), size, cols, rows }
     }
 
+    /** Get element assuming data are stored in column-major order */
+    export function get(m: Matrix, i: number, j: number) { return m.data[m.rows * j + i]; }
+    /** Set element assuming data are stored in column-major order */
+    export function set(m: Matrix, i: number, j: number, value: number) { m.data[m.rows * j + i] = value; }
+    /** Add to element assuming data are stored in column-major order */
+    export function add(m: Matrix, i: number, j: number, value: number) { m.data[m.rows * j + i] += value; }
+    /** Zero out the matrix */
+    export function makeZero(m: Matrix) {
+        for (let i = 0, _l = m.data.length; i < _l; i++) m.data[i] = 0.0;
+    }
+
     export function fromArray(data: NumberArray, cols: number, rows: number): Matrix {
         return { data, size: cols * rows, cols, rows }
     }