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Any rectangular $m\times n$ matrix ${\mathit {A}}$ can be decomposed to a product of an orthogonal matrix ${\mathit {Q}}$ and an upper (right) triangular matrix ${\mathit {R}}$ , as described in QR decomposition.

Task

Demonstrate the QR decomposition on the example matrix from the Wikipedia article:

A={\begin{pmatrix}12&-51&4\\6&167&-68\\-4&24&-41\end{pmatrix}}

and the usage for linear least squares problems on the example from Polynomial_regression. The method of Householder reflections should be used:

Method

Multiplying a given vector ${\mathit {a}}$ , for example the first column of matrix ${\mathit {A}}$ , with the Householder matrix ${\mathit {H}}$ , which is given as

H=I-{\frac {2}{u^{T}u}}uu^{T}

reflects ${\mathit {a}}$ about a plane given by its normal vector ${\mathit {u}}$ . When the normal vector of the plane ${\mathit {u}}$ is given as

u=a-\|a\|_{2}\;e_{1}

then the transformation reflects ${\mathit {a}}$ onto the first standard basis vector

e_{1}=[1\;0\;0\;...]^{T}

which means that all entries but the first become zero. To avoid numerical cancellation errors, we should take the opposite sign of $a_{1}$ :

u=a+{\textrm {sign}}(a_{1})\|a\|_{2}\;e_{1}

and normalize with respect to the first element:

v={\frac {u}{u_{1}}}

The equation for $H$ thus becomes:

H=I-{\frac {2}{v^{T}v}}vv^{T}

or, in another form

H=I-\beta vv^{T}

with

\beta ={\frac {2}{v^{T}v}}

Applying ${\mathit {H}}$ on ${\mathit {a}}$ then gives

H\;a=-{\textrm {sign}}(a_{1})\;\|a\|_{2}\;e_{1}

and applying ${\mathit {H}}$ on the matrix ${\mathit {A}}$ zeroes all subdiagonal elements of the first column:

H_{1}\;A={\begin{pmatrix}r_{11}&r_{12}&r_{13}\\0&*&*\\0&*&*\end{pmatrix}}

In the second step, the second column of ${\mathit {A}}$ , we want to zero all elements but the first two, which means that we have to calculate ${\mathit {H}}$ with the first column of the submatrix (denoted *), not on the whole second column of ${\mathit {A}}$ .

To get $H_{2}$ , we then embed the new ${\mathit {H}}$ into an $m\times n$ identity:

H_{2}={\begin{pmatrix}1&0&0\\0&H&\\0&&\end{pmatrix}}

This is how we can, column by column, remove all subdiagonal elements of ${\mathit {A}}$ and thus transform it into ${\mathit {R}}$ .

H_{n}\;...\;H_{3}H_{2}H_{1}A=R

The product of all the Householder matrices ${\mathit {H}}$ , for every column, in reverse order, will then yield the orthogonal matrix ${\mathit {Q}}$ .

H_{1}H_{2}H_{3}\;...\;H_{n}=Q

The QR decomposition should then be used to solve linear least squares (Multiple regression) problems ${\mathit {A}}x=b$ by solving

R\;x=Q^{T}\;b

When ${\mathit {R}}$ is not square, i.e. $m>n$ we have to cut off the ${\mathit {m}}-n$ zero padded bottom rows.

R={\begin{pmatrix}R_{1}\\0\end{pmatrix}}

and the same for the RHS:

Q^{T}\;b={\begin{pmatrix}q_{1}\\q_{2}\end{pmatrix}}

Finally, solve the square upper triangular system by back substitution:

R_{1}\;x=q_{1}

Ada

Output matches that of Matlab solution, not tested with other matrices. <lang Ada> with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays; with Ada.Numerics.Generic_Elementary_Functions; procedure QR is

  procedure Show (mat : Real_Matrix) is
     package FIO is new Ada.Text_IO.Float_IO (Float);
  begin
     for row in mat'Range (1) loop
        for col in mat'Range (2) loop
           FIO.Put (mat (row, col), Exp => 0, Aft => 4, Fore => 5);
        end loop;
        New_Line;
     end loop;
  end Show;

  function GetCol (mat : Real_Matrix; n : Integer) return Real_Matrix is
     column : Real_Matrix (mat'Range (1), 1 .. 1);
  begin
     for row in mat'Range (1) loop
        column (row, 1) := mat (row, n);
     end loop;
     return column;
  end GetCol;

  function Mag (mat : Real_Matrix) return Float is
     sum : Real_Matrix := Transpose (mat) * mat;
     package Math is new Ada.Numerics.Generic_Elementary_Functions
        (Float);
  begin
     return Math.Sqrt (sum (1, 1));
  end Mag;

  function eVect (col : Real_Matrix; n : Integer) return Real_Matrix is
     vect : Real_Matrix (col'Range (1), 1 .. 1);
  begin
     for row in col'Range (1) loop
        if row /= n then vect (row, 1) := 0.0;
        else vect (row, 1) := 1.0; end if;
     end loop;
     return vect;
  end eVect;

  function Identity (n : Integer) return Real_Matrix is
     mat : Real_Matrix (1 .. n, 1 .. n) := (1 .. n => (others => 0.0));
  begin
     for i in Integer range 1 .. n loop mat (i, i) := 1.0; end loop;
     return mat;
  end Identity;

  function Chop (mat : Real_Matrix; n : Integer) return Real_Matrix is
     small : Real_Matrix (n .. mat'Length (1), n .. mat'Length (2));
  begin
     for row in small'Range (1) loop
        for col in small'Range (2) loop
           small (row, col) := mat (row, col);
        end loop;
     end loop;
     return small;
  end Chop;

  function H_n (inmat : Real_Matrix; n : Integer)
     return Real_Matrix is
     mat : Real_Matrix := Chop (inmat, n);
     col : Real_Matrix := GetCol (mat, n);
     colT : Real_Matrix (1 .. 1, mat'Range (1));
     H : Real_Matrix := Identity (mat'Length (1));
     Hall : Real_Matrix := Identity (inmat'Length (1));
  begin
     col := col - Mag (col) * eVect (col, n);
     col := col / Mag (col);
     colT := Transpose (col);
     H := H - 2.0 * (col * colT);
     for row in H'Range (1) loop
        for col in H'Range (2) loop
           Hall (n - 1 + row, n - 1 + col) := H (row, col);
        end loop;
     end loop;
     return Hall;
  end H_n;

  A : constant Real_Matrix (1 .. 3, 1 .. 3) := (
     (12.0, -51.0, 4.0),
     (6.0, 167.0, -68.0),
     (-4.0, 24.0, -41.0));
  Q1, Q2, Q3, Q, R: Real_Matrix (1 .. 3, 1 .. 3);

begin

  Q1 := H_n (A, 1);
  Q2 := H_n (Q1 * A, 2);
  Q3 := H_n (Q2 * Q1* A, 3);
  Q := Transpose (Q1) * Transpose (Q2) * TransPose(Q3);
  R := Q3 * Q2 * Q1 * A;
  Put_Line ("Q:"); Show (Q);
  Put_Line ("R:"); Show (R);

end QR;</lang>

Output:

Q:
    0.8571   -0.3943   -0.3314
    0.4286    0.9029    0.0343
   -0.2857    0.1714   -0.9429
R:
   14.0000   21.0000  -14.0000
   -0.0000  175.0000  -70.0000
   -0.0000    0.0000   35.0000

Axiom

The following provides a generic QR decomposition for arbitrary precision floats, double floats and exact calculations: <lang Axiom>)abbrev package TESTP TestPackage TestPackage(R:Join(Field,RadicalCategory)): with

   unitVector: NonNegativeInteger -> Vector(R)
   "/": (Vector(R),R) -> Vector(R)
   "^": (Vector(R),NonNegativeInteger) -> Vector(R)
   solveUpperTriangular: (Matrix(R),Vector(R)) -> Vector(R)
   signValue: R -> R
   householder: Vector(R) -> Matrix(R)
   qr: Matrix(R) -> Record(q:Matrix(R),r:Matrix(R))
   lsqr: (Matrix(R),Vector(R)) -> Vector(R)
   polyfit: (Vector(R),Vector(R),NonNegativeInteger) -> Vector(R)
 == add
   unitVector(dim) ==
     out := new(dim,0@R)$Vector(R)
     out(1) := 1@R
     out
   v:Vector(R) / a:R == map((vi:R):R +-> vi/a, v)$Vector(R)
   v:Vector(R) ^ n:NonNegativeInteger == map((vi:R):R +-> vi^n, v)$Vector(R)
   solveUpperTriangular(r,b) ==
     n := ncols r
     x := new(n,0@R)$Vector(R)
     for k in n..1 by -1 repeat
       index := min(n,k+1)

x(k) := (b(k)-reduce("+",subMatrix(r,k,k,index,n)*x.(index..n)))/r(k,k)

     x
   signValue(r) ==
     R has (sign: R -> Integer) => coerce(sign(r)$R)$R
     zero? r => r
     if sqrt(r*r) = r then 1 else -1
   householder(a) ==
     m := #a
     u := a + length(a)*signValue(a(1))*unitVector(m) 
     v := u/u(1) 
     beta := (1+1)/dot(v,v)
     scalarMatrix(m,1) - beta*transpose(outerProduct(v,v))
   qr(a) ==
     (m,n) := (nrows a, ncols a)
     qm := scalarMatrix(m,1)
     rm := copy a
     for i in 1..(if m=n then n-1 else n) repeat
       x := column(subMatrix(rm,i,m,i,i),1)

h := scalarMatrix(m,1) setsubMatrix!(h,i,i,householder x) qm := qm*h rm := h*rm

     [qm,rm]
   lsqr(a,b) ==
     dc := qr a
     n := ncols(dc.r)
     solveUpperTriangular(subMatrix(dc.r,1,n,1,n),transpose(dc.q)*b)
   polyfit(x,y,n) ==
     a := new(#x,n+1,0@R)$Matrix(R)
     for j in 0..n repeat
       setColumn!(a,j+1,x^j)
     lsqr(a,y)</lang>

This can be called using: <lang Axiom>m := matrix [[12, -51, 4], [6, 167, -68], [-4, 24, -41]]; qr m x := vector [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]; y := vector [1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]; polyfit(x, y, 2)</lang> With output in exact form: <lang Axiom>qr m

            +  6    69     58 +
            |- -   ---    --- |
            |  7   175    175 |
            |                 |    +- 14  - 21    14 +
            |  3    158     6 |    |                 |
        [q= |- -  - ---  - ---|,r= | 0    - 175   70 |]
            |  7    175    175|    |                 |
            |                 |    + 0      0    - 35+
            | 2      6    33  |
            | -   - --    --  |
            + 7     35    35  +

         Type: Record(q: Matrix(AlgebraicNumber),r: Matrix(AlgebraicNumber))

polyfit(x, y, 2)

  [1,2,3]
                             Type: Vector(AlgebraicNumber)</lang>

The calculations are comparable to those from the default QR decomposition in R.

BBC BASIC

Works with: BBC BASIC for Windows

Makes heavy use of BBC BASIC's matrix arithmetic. <lang bbcbasic> *FLOAT 64

     @% = &2040A
     INSTALL @lib$+"ARRAYLIB"
     
     REM Test matrix for QR decomposition:
     DIM A(2,2)
     A() = 12, -51,   4, \
     \      6, 167, -68, \
     \     -4,  24, -41
     
     REM Do the QR decomposition:
     DIM Q(2,2), R(2,2)
     PROCqrdecompose(A(), Q(), R())
     PRINT "Q:"
     PRINT Q(0,0), Q(0,1), Q(0,2)
     PRINT Q(1,0), Q(1,1), Q(1,2)
     PRINT Q(2,0), Q(2,1), Q(2,2)
     PRINT "R:"
     PRINT R(0,0), R(0,1), R(0,2)
     PRINT R(1,0), R(1,1), R(1,2)
     PRINT R(2,0), R(2,1), R(2,2)
     
     REM Test data for least-squares solution:
     DIM x(10) : x() = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
     DIM y(10) : y() = 1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321
     
     REM Do the least-squares solution:
     DIM a(10,2), q(10,10), r(10,2), t(10,10), b(10), z(2)
     FOR i% = 0 TO 10
       FOR j% = 0 TO 2
         a(i%,j%) = x(i%) ^ j%
       NEXT
     NEXT
     PROCqrdecompose(a(), q(), r())
     PROC_transpose(q(),t())
     b() = t() . y()
     FOR k% = 2 TO 0 STEP -1
       s = 0
       IF k% < 2 THEN
         FOR j% = k%+1 TO 2
           s += r(k%,j%) * z(j%)
         NEXT
       ENDIF
       z(k%) = (b(k%) - s) / r(k%,k%)
     NEXT k%
     PRINT '"Least-squares solution:"
     PRINT z(0), z(1), z(2)
     END
     
     DEF PROCqrdecompose(A(), Q(), R())
     LOCAL i%, k%, m%, n%, H()
     m% = DIM(A(),1) : n% = DIM(A(),2)
     DIM H(m%,m%)
     FOR i% = 0 TO m% : Q(i%,i%) = 1 : NEXT
     WHILE n%
       PROCqrstep(n%, k%, A(), H())
       A() = H() . A()
       Q() = Q() . H()
       k% += 1
       m% -= 1
       n% -= 1
     ENDWHILE
     R() = A()
     ENDPROC
     
     DEF PROCqrstep(n%, k%, A(), H())
     LOCAL a(), h(), i%, j%
     DIM a(n%,0), h(n%,n%)
     FOR i% = 0 TO n% : a(i%,0) = A(i%+k%,k%) : NEXT
     PROChouseholder(h(), a())
     H() = 0  : H(0,0) = 1
     FOR i% = 0 TO n%
       FOR j% = 0 TO n%
         H(i%+k%,j%+k%) = h(i%,j%)
       NEXT
     NEXT
     ENDPROC
     
     REM Create the Householder matrix for the supplied column vector:
     DEF PROChouseholder(H(), a())
     LOCAL e(), u(), v(), vt(), vvt(), I(), d()
     LOCAL i%, n% : n% = DIM(a(),1)
     REM Create the scaled standard basis vector e():
     DIM e(n%,0) : e(0,0) = SGN(a(0,0)) * MOD(a())
     REM Create the normal vector u():
     DIM u(n%,0) : u() = a() + e()
     REM Normalise with respect to the first element:
     DIM v(n%,0) : v() = u() / u(0,0)
     REM Get the transpose of v() and its dot product with v():
     DIM vt(0,n%), d(0) : PROC_transpose(v(), vt()) : d() = vt() . v()
     REM Get the product of v() and vt():
     DIM vvt(n%,n%) : vvt() = v() . vt()
     REM Create an identity matrix I():
     DIM I(n%,n%) : FOR i% = 0 TO n% : I(i%,i%) = 1 : NEXT
     REM Create the Householder matrix H() = I - 2/vt()v() v()vt():
     vvt() *= 2 / d(0) : H() = I() - vvt()
     ENDPROC</lang>

Output:

Q:
   -0.8571    0.3943    0.3314
   -0.4286   -0.9029   -0.0343
    0.2857   -0.1714    0.9429
R:
  -14.0000  -21.0000   14.0000
    0.0000 -175.0000   70.0000
    0.0000    0.0000  -35.0000

Least-squares solution:
    1.0000    2.0000    3.0000

C

<lang 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, int n) { 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; }

/* c = a + b * s */ 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; }

/* m = I - v v^T */ 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; }

/* ||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); }

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

/* take c-th column of m, put in v */ 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(in, 5, 3); householder(x, &R, &Q);

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

// to show their product is the input matrix 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; }</lang>

Output:

Q
    0.846   -0.391    0.343    0.082    0.078
    0.423    0.904   -0.029    0.026    0.045
   -0.282    0.170    0.933   -0.047   -0.137
   -0.071    0.014   -0.001    0.980   -0.184
    0.141   -0.017   -0.106   -0.171   -0.969

R
   14.177   20.667  -13.402
   -0.000  175.043  -70.080
    0.000    0.000  -35.202
   -0.000   -0.000   -0.000
    0.000    0.000   -0.000

Q * R
   12.000  -51.000    4.000
    6.000  167.000  -68.000
   -4.000   24.000  -41.000
   -1.000    1.000   -0.000
    2.000   -0.000    3.000

Common Lisp

Uses the routines m+, m-, .*, ./ from Element-wise_operations, mmul from Matrix multiplication, mtp from Matrix transposition.

Helper functions: <lang lisp>(defun sign (x)

 (if (zerop x)
     x
     (/ x (abs x))))

(defun norm (x)

 (let ((len (car (array-dimensions x))))
   (sqrt (loop for i from 0 to (1- len) sum (expt (aref x i 0) 2)))))

(defun make-unit-vector (dim)

 (let ((vec (make-array `(,dim ,1) :initial-element 0.0d0)))
   (setf (aref vec 0 0) 1.0d0)
   vec))

Return a nxn identity matrix.

(defun eye (n)

 (let ((I (make-array `(,n ,n) :initial-element 0)))
   (loop for j from 0 to (- n 1) do
         (setf (aref I j j) 1))
   I))

(defun array-range (A ma mb na nb)

 (let* ((mm (1+ (- mb ma)))
        (nn (1+ (- nb na)))
        (B (make-array `(,mm ,nn) :initial-element 0.0d0)))

   (loop for i from 0 to (1- mm) do
        (loop for j from 0 to (1- nn) do
             (setf (aref B i j)
                   (aref A (+ ma i) (+ na j)))))
   B))

(defun rows (A) (car (array-dimensions A))) (defun cols (A) (cadr (array-dimensions A))) (defun mcol (A n) (array-range A 0 (1- (rows A)) n n)) (defun mrow (A n) (array-range A n n 0 (1- (cols A))))

(defun array-embed (A B row col)

 (let* ((ma (rows A))
        (na (cols A))
        (mb (rows B))
        (nb (cols B))
        (C  (make-array `(,ma ,na) :initial-element 0.0d0)))

   (loop for i from 0 to (1- ma) do
        (loop for j from 0 to (1- na) do
             (setf (aref C i j) (aref A i j))))

   (loop for i from 0 to (1- mb) do
        (loop for j from 0 to (1- nb) do
             (setf (aref C (+ row i) (+ col j))
                   (aref B i j))))

C))

</lang>

Main routines: <lang lisp> (defun make-householder (a)

 (let* ((m    (car (array-dimensions a)))
        (s    (sign (aref a 0 0)))
        (e    (make-unit-vector m))
        (u    (m+ a (.* (* (norm a) s) e)))
        (v    (./ u (aref u 0 0)))
        (beta (/ 2 (aref (mmul (mtp v) v) 0 0))))
   
   (m- (eye m)
       (.* beta (mmul v (mtp v))))))

(defun qr (A)

 (let* ((m (car  (array-dimensions A)))
        (n (cadr (array-dimensions A)))
        (Q (eye m)))

   ;; Work on n columns of A.
   (loop for i from 0 to (if (= m n) (- n 2) (- n 1)) do

        ;; Select the i-th submatrix. For i=0 this means the original matrix A.
        (let* ((B (array-range A i (1- m) i (1- n)))
               ;; Take the first column of the current submatrix B.
               (x (mcol B 0))
               ;; Create the Householder matrix for the column and embed it into an mxm identity.
               (H (array-embed (eye m) (make-householder x) i i)))

          ;; The product of all H matrices from the right hand side is the orthogonal matrix Q.
          (setf Q (mmul Q H))

          ;; The product of all H matrices with A from the LHS is the upper triangular matrix R.
          (setf A (mmul H A))))

   ;; Return Q and R.
   (values Q A)))

</lang>

Example 1:

<lang lisp>(qr #2A((12 -51 4) (6 167 -68) (-4 24 -41)))

2A((-0.85 0.39 0.33)

   (-0.42 -0.90 -0.03)
   ( 0.28 -0.17  0.94))

2A((-14.0 -21.0 14.0)

   (  0.0 -175.0  70.0)
   (  0.0    0.0 -35.0))</lang>

Example 2, Polynomial regression:

<lang lisp>(defun polyfit (x y n)

 (let* ((m (cadr (array-dimensions x)))
        (A (make-array `(,m ,(+ n 1)) :initial-element 0)))
   (loop for i from 0 to (- m 1) do
         (loop for j from 0 to n do
               (setf (aref A i j)
                     (expt (aref x 0 i) j))))
   (lsqr A (mtp y))))

Solve a linear least squares problem by QR decomposition.

(defun lsqr (A b)

 (multiple-value-bind (Q R) (qr A)
   (let* ((n (cadr (array-dimensions R))))
     (solve-upper-triangular (array-range R                0 (- n 1) 0 (- n 1))
                             (array-range (mmul (mtp Q) b) 0 (- n 1) 0 0)))))

Solve an upper triangular system by back substitution.

(defun solve-upper-triangular (R b)

 (let* ((n (cadr (array-dimensions R)))
        (x (make-array `(,n 1) :initial-element 0.0d0)))

   (loop for k from (- n 1) downto 0
      do (setf (aref x k 0)
               (/ (- (aref b k 0)
                     (loop for j from (+ k 1) to (- n 1)
                        sum (* (aref R k j)
                               (aref x j 0))))
                  (aref R k k))))
   x))</lang>

<lang lisp>;; Finally use the data: (let ((x #2A((0 1 2 3 4 5 6 7 8 9 10)))

     (y #2A((1 6 17 34 57 86 121 162 209 262 321))))
   (polyfit x y 2))

2A((0.999999966345088) (2.000000015144699) (2.99999999879804))</lang>

D

Translation of: Common Lisp

Uses the functions copied from Element-wise_operations, Matrix multiplication, and Matrix transposition. <lang d>import std.stdio, std.math, std.algorithm, std.traits,

      std.typecons, std.numeric, std.range, std.conv;

template elementwiseMat(string op) {

   T[][] elementwiseMat(T)(in T[][] A, in T B) pure nothrow {
       if (A.empty)
           return null;
       auto R = new typeof(return)(A.length, A[0].length);
       foreach (immutable r, const row; A)
           R[r][] = mixin("row[] " ~ op ~ "B");
       return R;
   }

   T[][] elementwiseMat(T, U)(in T[][] A, in U[][] B)
   pure nothrow if (is(Unqual!T == Unqual!U)) {
       assert(A.length == B.length);
       if (A.empty)
           return null;
       auto R = new typeof(return)(A.length, A[0].length);
       foreach (immutable r, const row; A) {
           assert(row.length == B[r].length);
           R[r][] = mixin("row[] " ~ op ~ "B[r][]");
       }
       return R;
   }

}

alias mSum = elementwiseMat!q{ + },

     mSub = elementwiseMat!q{ - },
     pMul = elementwiseMat!q{ * },
     pDiv = elementwiseMat!q{ / };

bool isRectangular(T)(in T[][] mat) pure nothrow {

   return mat.all!(r => r.length == mat[0].length);

}

T[][] matMul(T)(in T[][] a, in T[][] b) pure nothrow in {

   assert(a.isRectangular && b.isRectangular &&
          a[0].length == b.length);

} body {

   auto result = new T[][](a.length, b[0].length);
   auto aux = new T[b.length];
   foreach (immutable j; 0 .. b[0].length) {
       foreach (immutable k; 0 .. b.length)
           aux[k] = b[k][j];
       foreach (immutable i; 0 .. a.length)
           result[i][j] = a[i].dotProduct(aux);
   }
   return result;

}

Unqual!T[][] transpose(T)(in T[][] m) pure nothrow {

   auto r = new Unqual!T[][](m[0].length, m.length);
   foreach (immutable nr, row; m)
       foreach (immutable nc, immutable c; row)
           r[nc][nr] = c;
   return r;

}

T norm(T)(in T[][] m) pure nothrow {

   return transversal(m, 0).map!q{ a ^^ 2 }.sum.sqrt;

}

Unqual!T[][] makeUnitVector(T)(in size_t dim) pure nothrow {

   auto result = new Unqual!T[][](dim, 1);
   foreach (row; result)
       row[] = 0;
   result[0][0] = 1;
   return result;

}

/// Return a nxn identity matrix. Unqual!T[][] matId(T)(in size_t n) pure nothrow {

   auto Id = new Unqual!T[][](n, n);
   foreach (immutable r, row; Id) {
       row[] = 0;
       row[r] = 1;
   }
   return Id;

}

T[][] slice2D(T)(in T[][] A,

                in size_t ma, in size_t mb,
                in size_t na, in size_t nb) pure nothrow {
   auto B = new T[][](mb - ma + 1, nb - na + 1);
   foreach (immutable i, brow; B)
       brow[] = A[ma + i][na .. na + brow.length];
   return B;

}

size_t rows(T)(in T[][] A) pure nothrow { return A.length; }

size_t cols(T)(in T[][] A) pure nothrow {

   return A.length ? A[0].length : 0;

}

T[][] mcol(T)(in T[][] A, in size_t n) pure nothrow {

   return slice2D(A, 0, A.rows - 1, n, n);

}

T[][] matEmbed(T)(in T[][] A, in T[][] B,

                 in size_t row, in size_t col) pure nothrow {
   auto C = new T[][](rows(A), cols(A));
   foreach (immutable i, const arow; A)
       C[i][] = arow[]; // Some wasted copies.
   foreach (immutable i, const brow; B)
       C[row + i][col .. col + brow.length] = brow[];
   return C;

}

// Main routines ---------------

T[][] makeHouseholder(T)(in T[][] a) {

   immutable m = a.rows;
   immutable T s = a[0][0].sgn;
   immutable e = makeUnitVector!T(m);
   immutable u = mSum(a, pMul(e, a.norm * s));
   immutable v = pDiv(u, u[0][0]);
   immutable beta = 2.0 / v.transpose.matMul(v)[0][0];
   return mSub(matId!T(m), pMul(v.matMul(v.transpose), beta));

}

Tuple!(T[][],"Q", T[][],"R") QRdecomposition(T)(T[][] A) {

   immutable m = A.rows;
   immutable n = A.cols;
   auto Q = matId!T(m);

   // Work on n columns of A.
   foreach (immutable i; 0 .. (m == n ? n - 1 : n)) {
       // Select the i-th submatrix. For i=0 this means the original
       // matrix A.
       immutable B = slice2D(A, i, m - 1, i, n - 1);

       // Take the first column of the current submatrix B.
       immutable x = mcol(B, 0);

       // Create the Householder matrix for the column and embed it
       // into an mxm identity.
       immutable H = matEmbed(matId!T(m), x.makeHouseholder, i, i);

       // The product of all H matrices from the right hand side is
       // the orthogonal matrix Q.
       Q = Q.matMul(H);

       // The product of all H matrices with A from the LHS is the
       // upper triangular matrix R.
       A  = H.matMul(A);
   }

   // Return Q and R.
   return typeof(return)(Q, A);

}

// Polynomial regression ---------------

/// Solve an upper triangular system by back substitution. T[][] solveUpperTriangular(T)(in T[][] R, in T[][] b) pure nothrow {

   immutable n = R.cols;
   auto x = new T[][](n, 1);

   foreach_reverse (immutable k; 0 .. n) {
       T tot = 0;
       foreach (immutable j; k + 1 .. n)
           tot += R[k][j] * x[j][0];
       x[k][0] = (b[k][0] - tot) / R[k][k];
   }

   return x;

}

/// Solve a linear least squares problem by QR decomposition. T[][] lsqr(T)(T[][] A, in T[][] b) pure nothrow {

   const qr = A.QRdecomposition;
   immutable n = qr.R.cols;
   return solveUpperTriangular(
       slice2D(qr.R, 0, n - 1, 0, n - 1),
       slice2D(qr.Q.transpose.matMul(b), 0, n - 1, 0, 0));

}

T[][] polyFit(T)(in T[][] x, in T[][] y, in size_t n) pure nothrow {

   immutable size_t m = x.cols;
   auto A = new T[][](m, n + 1);
   foreach (immutable i, row; A)
       foreach (immutable j, ref item; row)
           item = x[0][i] ^^ j;
   return lsqr(A, y.transpose);

}

void main() {

   // immutable (Q, R) = QRdecomposition([[12.0, -51,   4],
   immutable qr = QRdecomposition([[12.0, -51,   4],
                                   [ 6.0, 167, -68],
                                   [-4.0,  24, -41]]);
   immutable form = "[%([%(%2.3f, %)]%|,\n %)]\n";
   writefln(form, qr.Q);
   writefln(form, qr.R);

   immutable x = 0.0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
   immutable y = 1.0, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321;
   polyFit(x, y, 2).writeln;

}</lang>

Output:

[[-0.857, 0.394, 0.331],
 [-0.429, -0.903, -0.034],
 [0.286, -0.171, 0.943]]

[[-14.000, -21.000, 14.000],
 [0.000, -175.000, 70.000],
 [0.000, -0.000, -35.000]]

[[1], [2], [3]]

Go

Translation of: Common Lisp

A fairly close port of the Common Lisp solution, this solution uses the go.matrix library for supporting functions. Note though, that go.matrix has QR decomposition, as shown in the Go solution to Polynomial regression. The solution there is coded more directly than by following the CL example here. Similarly, examination of the go.matrix QR source shows that it computes the decomposition more directly. <lang 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))
   // (error checking skipped in this solution)
   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++ {
       // (copy is only for compatibility with an older version of gomatrix)
       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() {

   // task 1: show qr decomp of wp example
   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)

   // task 2: use qr decomp for polynomial regression example
   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

}</lang> Output:

q:
 {-0.857143,  0.394286,  0.331429,
 -0.428571, -0.902857, -0.034286,
  0.285714, -0.171429,  0.942857}
r:
 { -14,  -21,   14,
    0, -175,   70,
    0,    0,  -35}

polyfit:
 {1,
 2,
 3}

J

Solution (built-in):<lang j> QR =: 128!:0</lang> Solution (custom implementation): <lang j> mp=: +/ . * NB. matrix product

  h =: +@|:    NB. conjugate transpose

  QR=: 3 : 0
   n=.{:$A=.y
   if. 1>:n do.
    A ((% {.@,) ; ]) %:(h A) mp A
   else.
    m =.>.n%2
    A0=.m{."1 A
    A1=.m}."1 A
    'Q0 R0'=.QR A0
    'Q1 R1'=.QR A1 - Q0 mp T=.(h Q0) mp A1
    (Q0,.Q1);(R0,.T),(-n){."1 R1
   end.
  )</lang>

Example: <lang j> QR 12 _51 4,6 167 _68,:_4 24 _41 +-----------------------------+----------+ | 0.857143 _0.394286 _0.331429|14 21 _14| | 0.428571 0.902857 0.0342857| 0 175 _70| |_0.285714 0.171429 _0.942857| 0 0 35| +-----------------------------+----------+</lang>

Example (polynomial fitting using QR reduction):<lang j> X=:i.# Y=:1 6 17 34 57 86 121 162 209 262 321

  'Q R'=: QR X ^/ i.3
  R %.~(|:Q)+/ .* Y

1 2 3</lang> Notes:J offers a built-in QR decomposition function, 128!:0. If J did not offer this function as a built-in, it could be written in J along the lines of the second version, which is covered in an essay on the J wiki.

Julia

Built-in function <lang julia>Q, R = qr([12 -51 4; 6 167 -68; -4 24 -41])</lang>

Output:

(
3x3 Array{Float64,2}:
 -0.857143   0.394286   0.331429 
 -0.428571  -0.902857  -0.0342857
  0.285714  -0.171429   0.942857 ,

3x3 Array{Float64,2}:
 -14.0   -21.0   14.0
   0.0  -175.0   70.0
   0.0     0.0  -35.0)

Maple

<lang Maple> with(LinearAlgebra):

Q,R := QRDecomposition( evalf( <<12|-51|4>,<6|167|-68>,<-4|24|-41>>) ):

Q; R; </lang> Output:

        [-0.857142857142857   0.394285714285714    0.331428571428571]
        [                                                           ]
        [-0.428571428571429  -0.902857142857143  -0.0342857142857143]
        [                                                           ]
        [ 0.285714285714286  -0.171428571428571    0.942857142857143]

                 [-14.               -21.  14.0000000000000]
                 [                                         ]
                 [  0.  -175.000000000000  70.0000000000000]
                 [                                         ]
                 [  0.                 0.              -35.]

Mathematica

<lang Mathematica>{q,r}=QRDecomposition[{{12, -51, 4}, {6, 167, -68}, {-4, 24, -41}}]; q//MatrixForm

-> 6/7 3/7 -(2/7) -69/175 158/175 6/35 -58/175 6/175 -33/35

r//MatrixForm -> 14 21 -14

  0  175 -70
  0  0  35</lang>

MATLAB / Octave

<lang Matlab> A = [12 -51 4

      6 167 -68
     -4  24 -41];
[Q,R]=qr(A) </lang>

Output:

Q =

   0.857143  -0.394286  -0.331429
   0.428571   0.902857   0.034286
  -0.285714   0.171429  -0.942857

R =

    14    21   -14
     0   175   -70
     0     0    35

Maxima

<lang maxima>load(lapack)$ /* This may hang up in wxMaxima, if this happens, use xMaxima or plain MAxima in a terminal */

a: matrix([12, -51, 4],

         [ 6, 167, -68],
         [-4,  24, -41])$

[q, r]: dgeqrf(a)$

mat_norm(q . r - a, 1); 4.2632564145606011E-14

/* Note: the lapack package is a lisp translation of the fortran lapack library */</lang> For an exact or arbitrary precision solution:<lang maxima>load("linearalgebra")$ load("eigen")$ unitVector(n) := ematrix(n,1,1,1,1); signValue(r) := block([s:sign(r)],

 if s='pos then 1 else if s='zero then 0 else -1);

householder(a) := block([m : length(a),u,v,beta],

 u : a + sqrt(a .  a)*signValue(a[1,1])*unitVector(m),
 v : u / u[1,1],
 beta : 2/(v . v),
 diagmatrix(m,1) - beta*transpose(v . transpose(v)));

getSubmatrix(obj,i1,j1,i2,j2) := genmatrix(lambda([i,j], obj[i+i1-1,j+j1-1]),i2-i1+1,j2-j1+1); setSubmatrix(obj,i1,j1,subobj) := block([m,n],

 [m,n] : matrix_size(subobj),
 for i: 0 thru m-1 do
 (for j: 0 thru n-1 do
   obj[i1+i,j1+j] : subobj[i+1,j+1]));

qr(obj) := block([m,n,qm,rm,i],

 [m,n] : matrix_size(obj),
 qm : diagmatrix(m,1),
 rm : copymatrix(obj),
 for i: 1 thru (if m=n then n-1 else n) do
 block([x,h],
   x : getSubmatrix(rm,i,i,m,i),
   h : diagmatrix(m,1),
   setSubmatrix(h,i,i,householder(x)),
   qm : qm . h,
   rm : h . rm),
 [qm,rm]);

solveUpperTriangular(r,b) := block([n,x,index,k],

 n : second(matrix_size(r)),
 x : genmatrix(lambda([a, b], 0), n, 1),
 for k: n thru 1 step -1 do
 (index : min(n,k+1),
   x[k,1] : (b[k,1] - (getSubmatrix(r,k,index,k,n) . getSubmatrix(x,index,1,n,1)))/r[k,k]),
 x);

lsqr(a,b) := block([q,r,n],

 [q,r] : qr(a),
 n : second(matrix_size(r)),
 solveUpperTriangular(getSubmatrix(r,1,1,n,n), transpose(q) . b));

polyfit(x,y,n) := block([a,j],

 a : genmatrix(lambda([i,j], if j=1 then 1.0b0 else bfloat(x[i,1]^(j-1))),
   length(x),n+1),
 lsqr(a,y));</lang>Then we have the examples:<lang maxima>(%i) [q,r] : qr(a);

                [   6   69     58   ]
                [ - -   ---    ---  ]
                [   7   175    175  ]
                [                   ]  [ - 14  - 21    14  ]
                [   3    158     6  ]  [                   ]

(%o) [[ - - - --- - --- ], [ 0 - 175 70 ]]

                [   7    175    175 ]  [                   ]
                [                   ]  [  0      0    - 35 ]
                [  2     6     33   ]
                [  -   - --    --   ]
                [  7     35    35   ]

(%i) mat_norm(q . r - a, 1);

(%o) 0 (%i) x : transpose(matrix([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]))$

(%i) y : transpose(matrix([1, 6, 17, 34, 57, 86, 121, 162, 209, 262, 321]))$

(%i) fpprec : 30$

(%i) polyfit(x, y, 2);

                   [ 9.99999999999999999999999999996b-1 ]
                   [                                    ]

(%o) [ 2.00000000000000000000000000002b0 ]

                   [                                    ]
                   [               3.0b0                ]</lang>

PARI/GP

Works with: PARI/GP version 2.6.0 and above

<lang parigp>matqr(M)</lang>

R

<lang r># R has QR decomposition built-in (using LAPACK or LINPACK)

a <- matrix(c(12, -51, 4, 6, 167, -68, -4, 24, -41), nrow=3, ncol=3, byrow=T) d <- qr(a) qr.Q(d) qr.R(d)

now fitting a polynomial

x <- 0:10 y <- 3*x^2 + 2*x + 1

using QR decomposition directly

a <- cbind(1, x, x^2) qr.coef(qr(a), y)

using least squares

a <- cbind(x, x^2) lsfit(a, y)$coefficients

using a linear model

xx <- x*x m <- lm(y ~ x + xx) coef(m)</lang>

Racket

Racket has QR-decomposition builtin: <lang racket> > (require math) > (matrix-qr (matrix [[12 -51 4]

                     [ 6 167 -68]
                     [-4  24 -41]]))

(array #[#[6/7 -69/175 -58/175] #[3/7 158/175 6/175] #[-2/7 6/35 -33/35]]) (array #[#[14 21 -14] #[0 175 -70] #[0 0 35]]) </lang>

The builtin QR-decomposition uses the Gram-Schmidt algorithm.

Here is an implementation of the Householder method: <lang racket>

lang racket

(require math/matrix math/array) (define-values (T I col size)

 (values ; short names
  matrix-transpose identity-matrix matrix-col matrix-num-rows))

(define (scale c A) (matrix-scale A c)) (define (unit n i) (build-matrix n 1 (λ (j _) (if (= j i) 1 0))))

(define (H u)

 (matrix- (I (size u))
          (scale (/ 2 (matrix-dot u u))
                 (matrix* u (T u)))))

(define (normal a)

 (define a0 (matrix-ref a 0 0))
 (matrix- a (scale (* (sgn a0) (matrix-2norm a))
                   (unit (size a) 0))))

(define (QR A)

 (define n (size A))
 (for/fold ([Q (I n)] [R A]) ([i (- n 1)])
   (define Hi (H (normal (submatrix R (:: i n) (:: i (+ i 1))))))
   (define Hi* (if (= i 0) Hi (block-diagonal-matrix (list (I i) Hi)))) 
   (values (matrix* Q Hi*) (matrix* Hi* R))))

(QR (matrix [[12 -51 4]

              [ 6 167 -68]
              [-4  24 -41]]))

</lang> Output: <lang racket> (array #[#[6/7 69/175 -58/175]

            #[3/7 -158/175 6/175] 
            #[-2/7 -6/35 -33/35]])

(array #[#[14 21 -14]

            #[0 -175 70] 
            #[0 0 35]])

</lang>

Rascal

This function applies the Gram Schmidt algorithm. Q is printed in the console, R can be printed or visualized.

<lang Rascal>import util::Math; import Prelude; import vis::Figure; import vis::Render;

public rel[real,real,real] QRdecomposition(rel[real x, real y, real v] matrix){ //orthogonalcolumns oc = domainR(matrix, {0.0}); for (x <- sort(toList(domain(matrix)-{0.0}))){ c = domainR(matrix, {x}); o = domainR(oc, {x-1});

for (n <- [1.0 .. x]){ o = domainR(oc, {n-1}); c = matrixSubtract(c, matrixMultiplybyN(o, matrixDotproduct(o, c)/matrixDotproduct(o, o))); }

oc += c; }

Q = {}; //from orthogonal to orthonormal columns for (el <- oc){ c = domainR(oc, {el[0]}); Q += matrixNormalize({el}, c); }

//from Q to R R= matrixMultiplication(matrixTranspose(Q), matrix); R= {<x,y,toReal(round(v))> | <x,y,v> <- R};

println("Q:"); iprintlnExp(Q); println(); println("R:"); return R; }

//a function that takes the transpose of a matrix, see also Rosetta Code problem "Matrix transposition" public rel[real, real, real] matrixTranspose(rel[real x, real y, real v] matrix){ return {<y, x, v> | <x, y, v> <- matrix}; }

//a function to normalize an element of a matrix by the normalization of a column public rel[real,real,real] matrixNormalize(rel[real x, real y, real v] element, rel[real x, real y, real v] column){ normalized = 1.0/nroot((0.0 | it + v*v | <x,y,v> <- column), 2); return matrixMultiplybyN(element, normalized); }

//a function that takes the dot product, see also Rosetta Code problem "Dot product" public real matrixDotproduct(rel[real x, real y, real v] column1, rel[real x, real y, real v] column2){ return (0.0 | it + v1*v2 | <x1,y1,v1> <- column1, <x2,y2,v2> <- column2, y1==y2); }

//a function to subtract two columns public rel[real,real,real] matrixSubtract(rel[real x, real y, real v] column1, rel[real x, real y, real v] column2){ return {<x1,y1,v1-v2> | <x1,y1,v1> <- column1, <x2,y2,v2> <- column2, y1==y2}; }

//a function to multiply a column by a number public rel[real,real,real] matrixMultiplybyN(rel[real x, real y, real v] column, real n){ return {<x,y,v*n> | <x,y,v> <- column}; }

//a function to perform matrix multiplication, see also Rosetta Code problem "Matrix multiplication". public rel[real, real, real] matrixMultiplication(rel[real x, real y, real v] matrix1, rel[real x, real y, real v] matrix2){ if (max(matrix1.x) == max(matrix2.y)){ p = {<x1,y1,x2,y2, v1*v2> | <x1,y1,v1> <- matrix1, <x2,y2,v2> <- matrix2};

result = {}; for (y <- matrix1.y){ for (x <- matrix2.x){ v = (0.0 | it + v | <x1, y1, x2, y2, v> <- p, x==x2 && y==y1, x1==y2 && y2==x1); result += <x,y,v>; } } return result; } else throw "Matrix sizes do not match."; }

// a function to visualize the result public void displayMatrix(rel[real x, real y, real v] matrix){ points = [box(text("<v>"), align(0.3333*(x+1),0.3333*(y+1)),shrink(0.25)) | <x,y,v> <- matrix]; render(overlay([*points], aspectRatio(1.0))); }

//a matrix, given by a relation of <x-coordinate, y-coordinate, value>. public rel[real x, real y, real v] matrixA = { <0.0,0.0,12.0>, <0.0,1.0, 6.0>, <0.0,2.0,-4.0>, <1.0,0.0,-51.0>, <1.0,1.0,167.0>, <1.0,2.0,24.0>, <2.0,0.0,4.0>, <2.0,1.0,-68.0>, <2.0,2.0,-41.0> };</lang>

Example using visualization

rascal>displayMatrix(QRdecomposition(matrixA))

Q:
{
  <1.0,0.0,-0.394285714285714285714285714285714285714285714285714285714285714285713300>,
  <0.0,0.0,0.857142857142857142857142857142857142857142857142857142857142857142840>,
  <0.0,1.0,0.428571428571428571428571428571428571428571428571428571428571428571420>,
  <0.0,2.0,-0.285714285714285714285714285714285714285714285714285714285714285714280>,
  <2.0,0.0,-0.33142857142857142857142857142857142857142857142857142857142857142858800>,
  <1.0,2.0,0.171428571428571428571428571428571428571428571428571428571428571428571000>,
  <2.0,2.0,-0.94285714285714285714285714285714285714285714285714285714285714285719000>,
  <1.0,1.0,0.902857142857142857142857142857142857142857142857142857142857142857140600>,
  <2.0,1.0,0.03428571428571428571428571428571428571428571428571428571428571428571600>
}
See R in picture

SAS

<lang sas>/* See http://support.sas.com/documentation/cdl/en/imlug/63541/HTML/default/viewer.htm#imlug_langref_sect229.htm */

proc iml; a={12 -51 4,6 167 -68,-4 24 -41}; print(a); call qr(q,r,p,d,a); print(q); print(r); quit;

/*

          12       -51         4
           6       167       -68
          -4        24       -41

   -0.857143 0.3942857 -0.331429
   -0.428571 -0.902857 0.0342857
   0.2857143 -0.171429 -0.942857

         -14       -21        14
           0      -175        70
           0         0        35

/</lang>

Tcl

Assuming the presence of the Tcl solutions to these tasks: Element-wise operations, Matrix multiplication, Matrix transposition

Translation of: Common Lisp

<lang tcl>package require Tcl 8.5 namespace path {::tcl::mathfunc ::tcl::mathop} proc sign x {expr {$x == 0 ? 0 : $x < 0 ? -1 : 1}} proc norm vec {

   set s 0
   foreach x $vec {set s [expr {$s + $x**2}]}
   return [sqrt $s]

} proc unitvec n {

   set v [lrepeat $n 0.0]
   lset v 0 1.0
   return $v

} proc I n {

   set m [lrepeat $n [lrepeat $n 0.0]]
   for {set i 0} {$i < $n} {incr i} {lset m $i $i 1.0}
   return $m

}

proc arrayEmbed {A B row col} {

   # $A will be copied automatically; Tcl values are copy-on-write
   lassign [size $B] mb nb
   for {set i 0} {$i < $mb} {incr i} {

for {set j 0} {$j < $nb} {incr j} { lset A [expr {$row + $i}] [expr {$col + $j}] [lindex $B $i $j] }

   }
   return $A

}

Unlike the Common Lisp version, here we use a specialist subcolumn
extraction function: like that, there's a lot less intermediate memory allocation
and the code is actually clearer.

proc subcolumn {A size column} {

   for {set i $column} {$i < $size} {incr i} {lappend x [lindex $A $i $column]}
   return $x

}

proc householder A {

   lassign [size $A] m
   set U [m+ $A [.* [unitvec $m] [expr {[norm $A] * [sign [lindex $A 0 0]]}]]]
   set V [./ $U [lindex $U 0 0]]
   set beta [expr {2.0 / [lindex [matrix_multiply [transpose $V] $V] 0 0]}]
   return [m- [I $m] [.* [matrix_multiply $V [transpose $V]] $beta]]

}

proc qrDecompose A {

   lassign [size $A] m n
   set Q [I $m]
   for {set i 0} {$i < ($m==$n ? $n-1 : $n)} {incr i} {

# Construct the Householder matrix set H [arrayEmbed [I $m] [householder [subcolumn $A $n $i]] $i $i] # Apply to build the decomposition set Q [matrix_multiply $Q $H] set A [matrix_multiply $H $A]

   }
   return [list $Q $A]

}</lang> Demonstrating: <lang tcl>set demo [qrDecompose {{12 -51 4} {6 167 -68} {-4 24 -41}}] puts "==Q==" print_matrix [lindex $demo 0] "%f" puts "==R==" print_matrix [lindex $demo 1] "%.1f"</lang> Output:

==Q==
-0.857143  0.394286  0.331429 
-0.428571 -0.902857 -0.034286 
 0.285714 -0.171429  0.942857 
==R==
-14.0  -21.0  14.0 
  0.0 -175.0  70.0 
  0.0    0.0 -35.0

@@ Line 1: / Line 1: @@
 {{task|Matrices}}[[Category:Mathematics]]
-Any rectangular <math>m \times n</math> matrix <math>\mathit A</math> can be decomposed to a product of a orthogonal matrix <math>\mathit Q</math> and a upper (right) triangular matrix <math>\mathit R</math>, as described in [[wp:QR decomposition|QR decomposition]].
+Any rectangular <math>m \times n</math> matrix <math>\mathit A</math> can be decomposed to a product of an orthogonal matrix <math>\mathit Q</math> and an upper (right) triangular matrix <math>\mathit R</math>, as described in [[wp:QR decomposition|QR decomposition]].
 '''Task'''