Task: QR decomposition
C: #include <stdio.h>
#include <stdlib.h>
#include <math.h>

typedef struct {
	int m, n;
	double ** v;
} mat_t, *mat;

mat matrix_new(int m, int n)
{
	mat x = malloc(sizeof(mat_t));
	x->v = malloc(sizeof(double*) * m);
	x->v[0] = calloc(sizeof(double), m * n);
	for (int i = 0; i < m; i++)
		x->v[i] = x->v[0] + n * i;
	x->m = m;
	x->n = n;
	return x;
}

void matrix_delete(mat m)
{
	free(m->v[0]);
	free(m->v);
	free(m);
}

void matrix_transpose(mat m)
{
	for (int i = 0; i < m->m; i++) {
		for (int j = 0; j < i; j++) {
			double t = m->v[i][j];
			m->v[i][j] = m->v[j][i];
			m->v[j][i] = t;
		}
	}
}

mat matrix_copy(int n, double a[][n], int m)
{
	mat x = matrix_new(m, n);
	for (int i = 0; i < m; i++)
		for (int j = 0; j < n; j++)
			x->v[i][j] = a[i][j];
	return x;
}

mat matrix_mul(mat x, mat y)
{
	if (x->n != y->m) return 0;
	mat r = matrix_new(x->m, y->n);
	for (int i = 0; i < x->m; i++)
		for (int j = 0; j < y->n; j++)
			for (int k = 0; k < x->n; k++)
				r->v[i][j] += x->v[i][k] * y->v[k][j];
	return r;
}

mat matrix_minor(mat x, int d)
{
	mat m = matrix_new(x->m, x->n);
	for (int i = 0; i < d; i++)
		m->v[i][i] = 1;
	for (int i = d; i < x->m; i++)
		for (int j = d; j < x->n; j++)
			m->v[i][j] = x->v[i][j];
	return m;
}


double *vmadd(double a[], double b[], double s, double c[], int n)
{
	for (int i = 0; i < n; i++)
		c[i] = a[i] + s * b[i];
	return c;
}


mat vmul(double v[], int n)
{
	mat x = matrix_new(n, n);
	for (int i = 0; i < n; i++)
		for (int j = 0; j < n; j++)
			x->v[i][j] = -2 *  v[i] * v[j];
	for (int i = 0; i < n; i++)
		x->v[i][i] += 1;

	return x;
}


double vnorm(double x[], int n)
{
	double sum = 0;
	for (int i = 0; i < n; i++) sum += x[i] * x[i];
	return sqrt(sum);
}


double* vdiv(double x[], double d, double y[], int n)
{
	for (int i = 0; i < n; i++) y[i] = x[i] / d;
	return y;
}


double* mcol(mat m, double *v, int c)
{
	for (int i = 0; i < m->m; i++)
		v[i] = m->v[i][c];
	return v;
}

void matrix_show(mat m)
{
	for(int i = 0; i < m->m; i++) {
		for (int j = 0; j < m->n; j++) {
			printf(" %8.3f", m->v[i][j]);
		}
		printf("\n");
	}
	printf("\n");
}

void householder(mat m, mat *R, mat *Q)
{
	mat q[m->m];
	mat z = m, z1;
	for (int k = 0; k < m->n && k < m->m - 1; k++) {
		double e[m->m], x[m->m], a;
		z1 = matrix_minor(z, k);
		if (z != m) matrix_delete(z);
		z = z1;

		mcol(z, x, k);
		a = vnorm(x, m->m);
		if (m->v[k][k] > 0) a = -a;

		for (int i = 0; i < m->m; i++)
			e[i] = (i == k) ? 1 : 0;

		vmadd(x, e, a, e, m->m);
		vdiv(e, vnorm(e, m->m), e, m->m);
		q[k] = vmul(e, m->m);
		z1 = matrix_mul(q[k], z);
		if (z != m) matrix_delete(z);
		z = z1;
	}
	matrix_delete(z);
	*Q = q[0];
	*R = matrix_mul(q[0], m);
	for (int i = 1; i < m->n && i < m->m - 1; i++) {
		z1 = matrix_mul(q[i], *Q);
		if (i > 1) matrix_delete(*Q);
		*Q = z1;
		matrix_delete(q[i]);
	}
	matrix_delete(q[0]);
	z = matrix_mul(*Q, m);
	matrix_delete(*R);
	*R = z;
	matrix_transpose(*Q);
}

double in[][3] = {
	{ 12, -51,   4},
	{  6, 167, -68},
	{ -4,  24, -41},
	{ -1, 1, 0},
	{ 2, 0, 3},
};

int main()
{
	mat R, Q;
	mat x = matrix_copy(3, in, 5);
	householder(x, &R, &Q);

	puts("Q"); matrix_show(Q);
	puts("R"); matrix_show(R);

	
	mat m = matrix_mul(Q, R);
	puts("Q * R"); matrix_show(m);

	matrix_delete(x);
	matrix_delete(R);
	matrix_delete(Q);
	matrix_delete(m);
	return 0;
}

Go: package main

import (
    "fmt"
    "math"

    "github.com/skelterjohn/go.matrix"
)

func sign(s float64) float64 {
    if s > 0 {
        return 1
    } else if s < 0 {
        return -1
    }
    return 0
}

func unitVector(n int) *matrix.DenseMatrix {
    vec := matrix.Zeros(n, 1)
    vec.Set(0, 0, 1)
    return vec
}

func householder(a *matrix.DenseMatrix) *matrix.DenseMatrix {
    m := a.Rows()
    s := sign(a.Get(0, 0))
    e := unitVector(m)
    u := matrix.Sum(a, matrix.Scaled(e, a.TwoNorm()*s))
    v := matrix.Scaled(u, 1/u.Get(0, 0))
    
    prod, _ := v.Transpose().TimesDense(v)
    β := 2 / prod.Get(0, 0)

    prod, _ = v.TimesDense(v.Transpose())
    return matrix.Difference(matrix.Eye(m), matrix.Scaled(prod, β))
}

func qr(a *matrix.DenseMatrix) (q, r *matrix.DenseMatrix) {
    m := a.Rows()
    n := a.Cols()
    q = matrix.Eye(m)

    last := n - 1
    if m == n {
        last--
    }
    for i := 0; i <= last; i++ {
        
        b := a.GetMatrix(i, i, m-i, n-i).Copy()
        x := b.GetColVector(0)
        h := matrix.Eye(m)
        h.SetMatrix(i, i, householder(x))
        q, _ = q.TimesDense(h)
        a, _ = h.TimesDense(a)
    }
    return q, a
}

func main() {
    
    a := matrix.MakeDenseMatrixStacked([][]float64{
        {12, -51, 4},
        {6, 167, -68},
        {-4, 24, -41}})
    q, r := qr(a)
    fmt.Println("q:\n", q)
    fmt.Println("r:\n", r)

    
    x := matrix.MakeDenseMatrixStacked([][]float64{
        {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}})
    y := matrix.MakeDenseMatrixStacked([][]float64{
        {1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321}})
    fmt.Println("\npolyfit:\n", polyfit(x, y, 2))
}

func polyfit(x, y *matrix.DenseMatrix, n int) *matrix.DenseMatrix {
    m := x.Cols()
    a := matrix.Zeros(m, n+1)
    for i := 0; i < m; i++ {
        for j := 0; j <= n; j++ {
            a.Set(i, j, math.Pow(x.Get(0, i), float64(j)))
        }
    }
    return lsqr(a, y.Transpose())
}

func lsqr(a, b *matrix.DenseMatrix) *matrix.DenseMatrix {
    q, r := qr(a)
    n := r.Cols()
    prod, _ := q.Transpose().TimesDense(b)
    return solveUT(r.GetMatrix(0, 0, n, n), prod.GetMatrix(0, 0, n, 1))
}

func solveUT(r, b *matrix.DenseMatrix) *matrix.DenseMatrix {
    n := r.Cols()
    x := matrix.Zeros(n, 1)
    for k := n - 1; k >= 0; k-- {
        sum := 0.
        for j := k + 1; j < n; j++ {
            sum += r.Get(k, j) * x.Get(j, 0)
        }
        x.Set(k, 0, (b.Get(k, 0)-sum)/r.Get(k, k))
    }
    return x
}