matrix_alg.cpp

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 *   Author: Siddharth Bharat Purohit <sidbpurohit@gmail.com>
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/**
 * @file matrix_alg.cpp
 *
 * Matrix algebra on raw arrays
 */

#include "matrix_alg.h"
#include <px4_platform_common/defines.h>

/*
 *    Does matrix multiplication of two regular/square matrices
 *
 *    @param     A,           Matrix A
 *    @param     B,           Matrix B
 *    @param     n,           dimemsion of square matrices
 *    @returns                multiplied matrix i.e. A*B
 */

float *mat_mul(float *A, float *B, uint8_t n)
{
    float *ret = new float[n * n];
    memset(ret, 0.0f, n * n * sizeof(float));

    for (uint8_t i = 0; i < n; i++) {
        for (uint8_t j = 0; j < n; j++) {
            for (uint8_t k = 0; k < n; k++) {
                ret[i * n + j] += A[i * n + k] * B[k * n + j];
            }
        }
    }

    return ret;
}

static inline void swap(float &a, float &b)
{
    float c;
    c = a;
    a = b;
    b = c;
}

/*
 *    calculates pivot matrix such that all the larger elements in the row are on diagonal
 *
 *    @param     A,           input matrix matrix
 *    @param     pivot
 *    @param     n,           dimenstion of square matrix
 *    @returns                false = matrix is Singular or non positive definite, true = matrix inversion successful
 */

static void mat_pivot(float *A, float *pivot, uint8_t n)
{
    for (uint8_t i = 0; i < n; i++) {
        for (uint8_t j = 0; j < n; j++) {
            pivot[i * n + j] = (i == j);
        }
    }

    for (uint8_t i = 0; i < n; i++) {
        uint8_t max_j = i;

        for (uint8_t j = i; j < n; j++) {
            if (fabsf(A[j * n + i]) > fabsf(A[max_j * n + i])) {
                max_j = j;
            }
        }

        if (max_j != i) {
            for (uint8_t k = 0; k < n; k++) {
                swap(pivot[i * n + k], pivot[max_j * n + k]);
            }
        }
    }
}

/*
 *    calculates matrix inverse of Lower trangular matrix using forward substitution
 *
 *    @param     L,           lower triangular matrix
 *    @param     out,         Output inverted lower triangular matrix
 *    @param     n,           dimension of matrix
 */

static void mat_forward_sub(float *L, float *out, uint8_t n)
{
    // Forward substitution solve LY = I
    for (int i = 0; i < n; i++) {
        out[i * n + i] = 1 / L[i * n + i];

        for (int j = i + 1; j < n; j++) {
            for (int k = i; k < j; k++) {
                out[j * n + i] -= L[j * n + k] * out[k * n + i];
            }

            out[j * n + i] /= L[j * n + j];
        }
    }
}

/*
 *    calculates matrix inverse of Upper trangular matrix using backward substitution
 *
 *    @param     U,           upper triangular matrix
 *    @param     out,         Output inverted upper triangular matrix
 *    @param     n,           dimension of matrix
 */

static void mat_back_sub(float *U, float *out, uint8_t n)
{
    // Backward Substitution solve UY = I
    for (int i = n - 1; i >= 0; i--) {
        out[i * n + i] = 1 / U[i * n + i];

        for (int j = i - 1; j >= 0; j--) {
            for (int k = i; k > j; k--) {
                out[j * n + i] -= U[j * n + k] * out[k * n + i];
            }

            out[j * n + i] /= U[j * n + j];
        }
    }
}

/*
 *    Decomposes square matrix into Lower and Upper triangular matrices such that
 *    A*P = L*U, where P is the pivot matrix
 *    ref: http://rosettacode.org/wiki/LU_decomposition
 *    @param     U,           upper triangular matrix
 *    @param     out,         Output inverted upper triangular matrix
 *    @param     n,           dimension of matrix
 */

static void mat_LU_decompose(float *A, float *L, float *U, float *P, uint8_t n)
{
    memset(L, 0, n * n * sizeof(float));
    memset(U, 0, n * n * sizeof(float));
    memset(P, 0, n * n * sizeof(float));
    mat_pivot(A, P, n);

    float *APrime = mat_mul(P, A, n);

    for (uint8_t i = 0; i < n; i++) {
        L[i * n + i] = 1;
    }

    for (uint8_t i = 0; i < n; i++) {
        for (uint8_t j = 0; j < n; j++) {
            if (j <= i) {
                U[j * n + i] = APrime[j * n + i];

                for (uint8_t k = 0; k < j; k++) {
                    U[j * n + i] -= L[j * n + k] * U[k * n + i];
                }
            }

            if (j >= i) {
                L[j * n + i] = APrime[j * n + i];

                for (uint8_t k = 0; k < i; k++) {
                    L[j * n + i] -= L[j * n + k] * U[k * n + i];
                }

                L[j * n + i] /= U[i * n + i];
            }
        }
    }

    delete[] APrime;
}

/*
 *    matrix inverse code for any square matrix using LU decomposition
 *    inv = inv(U)*inv(L)*P, where L and U are triagular matrices and P the pivot matrix
 *    ref: http://www.cl.cam.ac.uk/teaching/1314/NumMethods/supporting/mcmaster-kiruba-ludecomp.pdf
 *    @param     m,           input 4x4 matrix
 *    @param     inv,      Output inverted 4x4 matrix
 *    @param     n,           dimension of square matrix
 *    @returns                false = matrix is Singular, true = matrix inversion successful
 */
bool mat_inverse(float *A, float *inv, uint8_t n)
{
    float *L, *U, *P;
    bool ret = true;
    L = new float[n * n];
    U = new float[n * n];
    P = new float[n * n];
    mat_LU_decompose(A, L, U, P, n);

    float *L_inv = new float[n * n];
    float *U_inv = new float[n * n];

    memset(L_inv, 0, n * n * sizeof(float));
    mat_forward_sub(L, L_inv, n);

    memset(U_inv, 0, n * n * sizeof(float));
    mat_back_sub(U, U_inv, n);

    // decomposed matrices no longer required
    delete[] L;
    delete[] U;

    float *inv_unpivoted = mat_mul(U_inv, L_inv, n);
    float *inv_pivoted = mat_mul(inv_unpivoted, P, n);

    //check sanity of results
    for (uint8_t i = 0; i < n; i++) {
        for (uint8_t j = 0; j < n; j++) {
            if (!PX4_ISFINITE(inv_pivoted[i * n + j])) {
                ret = false;
            }
        }
    }

    memcpy(inv, inv_pivoted, n * n * sizeof(float));

    //free memory
    delete[] inv_pivoted;
    delete[] inv_unpivoted;
    delete[] P;
    delete[] U_inv;
    delete[] L_inv;
    return ret;
}

bool inverse4x4(float m[], float invOut[])
{
    float inv[16], det;
    uint8_t i;

    inv[0] = m[5]  * m[10] * m[15] -
         m[5]  * m[11] * m[14] -
         m[9]  * m[6]  * m[15] +
         m[9]  * m[7]  * m[14] +
         m[13] * m[6]  * m[11] -
         m[13] * m[7]  * m[10];

    inv[4] = -m[4]  * m[10] * m[15] +
         m[4]  * m[11] * m[14] +
         m[8]  * m[6]  * m[15] -
         m[8]  * m[7]  * m[14] -
         m[12] * m[6]  * m[11] +
         m[12] * m[7]  * m[10];

    inv[8] = m[4]  * m[9] * m[15] -
         m[4]  * m[11] * m[13] -
         m[8]  * m[5] * m[15] +
         m[8]  * m[7] * m[13] +
         m[12] * m[5] * m[11] -
         m[12] * m[7] * m[9];

    inv[12] = -m[4]  * m[9] * m[14] +
          m[4]  * m[10] * m[13] +
          m[8]  * m[5] * m[14] -
          m[8]  * m[6] * m[13] -
          m[12] * m[5] * m[10] +
          m[12] * m[6] * m[9];

    inv[1] = -m[1]  * m[10] * m[15] +
         m[1]  * m[11] * m[14] +
         m[9]  * m[2] * m[15] -
         m[9]  * m[3] * m[14] -
         m[13] * m[2] * m[11] +
         m[13] * m[3] * m[10];

    inv[5] = m[0]  * m[10] * m[15] -
         m[0]  * m[11] * m[14] -
         m[8]  * m[2] * m[15] +
         m[8]  * m[3] * m[14] +
         m[12] * m[2] * m[11] -
         m[12] * m[3] * m[10];

    inv[9] = -m[0]  * m[9] * m[15] +
         m[0]  * m[11] * m[13] +
         m[8]  * m[1] * m[15] -
         m[8]  * m[3] * m[13] -
         m[12] * m[1] * m[11] +
         m[12] * m[3] * m[9];

    inv[13] = m[0]  * m[9] * m[14] -
          m[0]  * m[10] * m[13] -
          m[8]  * m[1] * m[14] +
          m[8]  * m[2] * m[13] +
          m[12] * m[1] * m[10] -
          m[12] * m[2] * m[9];

    inv[2] = m[1]  * m[6] * m[15] -
         m[1]  * m[7] * m[14] -
         m[5]  * m[2] * m[15] +
         m[5]  * m[3] * m[14] +
         m[13] * m[2] * m[7] -
         m[13] * m[3] * m[6];

    inv[6] = -m[0]  * m[6] * m[15] +
         m[0]  * m[7] * m[14] +
         m[4]  * m[2] * m[15] -
         m[4]  * m[3] * m[14] -
         m[12] * m[2] * m[7] +
         m[12] * m[3] * m[6];

    inv[10] = m[0]  * m[5] * m[15] -
          m[0]  * m[7] * m[13] -
          m[4]  * m[1] * m[15] +
          m[4]  * m[3] * m[13] +
          m[12] * m[1] * m[7] -
          m[12] * m[3] * m[5];

    inv[14] = -m[0]  * m[5] * m[14] +
          m[0]  * m[6] * m[13] +
          m[4]  * m[1] * m[14] -
          m[4]  * m[2] * m[13] -
          m[12] * m[1] * m[6] +
          m[12] * m[2] * m[5];

    inv[3] = -m[1] * m[6] * m[11] +
         m[1] * m[7] * m[10] +
         m[5] * m[2] * m[11] -
         m[5] * m[3] * m[10] -
         m[9] * m[2] * m[7] +
         m[9] * m[3] * m[6];

    inv[7] = m[0] * m[6] * m[11] -
         m[0] * m[7] * m[10] -
         m[4] * m[2] * m[11] +
         m[4] * m[3] * m[10] +
         m[8] * m[2] * m[7] -
         m[8] * m[3] * m[6];

    inv[11] = -m[0] * m[5] * m[11] +
          m[0] * m[7] * m[9] +
          m[4] * m[1] * m[11] -
          m[4] * m[3] * m[9] -
          m[8] * m[1] * m[7] +
          m[8] * m[3] * m[5];

    inv[15] = m[0] * m[5] * m[10] -
          m[0] * m[6] * m[9] -
          m[4] * m[1] * m[10] +
          m[4] * m[2] * m[9] +
          m[8] * m[1] * m[6] -
          m[8] * m[2] * m[5];

    det = m[0] * inv[0] + m[1] * inv[4] + m[2] * inv[8] + m[3] * inv[12];

    if (fabsf(det) < 1.1755e-38f) {
        return false;
    }

    det = 1.0f / det;

    for (i = 0; i < 16; i++) {
        invOut[i] = inv[i] * det;
    }

    return true;
}