restart; with(LinearAlgebra);

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

D\303\251composition QR d'une matrice r\303\251elle

gs:=proc(a) local r,b,i,j,k,l,n,c,s,m; n:=ColumnDimension(a); m:=Vector(n,j->a[1..n,j]); r:=Matrix(n,n,0); for i from 1 to n do for j from 1 to i-1 do r[j,i]:=evalf(m[i].b[j]); od; c[i]:=m[i]; for j from 1 to i-1 do c[i]:=evalf(c[i]-r[j,i]*b[j]); od; r[i,i]:=evalf(VectorNorm(c[i],2)); b[i]:=c[i]/r[i,i]; od; [r,Matrix(n,n,(k,l)->b[l][k])]; end;

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

m:=<<1,2,3>|<1,0,1>|<0,2,1>>;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKXNUXTU=

dec:=gs(m);

NyQtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCIpIVFnNiItRiQ2Iy9GKyIob0dpIw==

%[2].%[1];

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKWMoXDgi

P:=Transpose(dec[2]).dec[2];

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKVNwbTU=

Matrix(3,3,(i,j)->round(P[i,j]));

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKEcoXEE=

E1:=proc(n) Vector(n,i->if i=1 then 1 else 0 fi) end;

Zio2I0kibkc2IkYlRiVGJS1JJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkOSRmKjYjSSJpR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiVAJS9GLCIiIkY1IiIhRiVGJUYlRiVGJUYl

E1(4);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpJT0/QSI=

La fonction suivante calcule la matrice de Householder d'un vecteur X donn\303\251. HH:=proc(X) local W,n; n:=Dimension(X); W:=evalf(X-Norm(X,2)*E1(n)); IdentityMatrix(n)-2/(Transpose(W).W)*W.Transpose(W) end;

Zio2I0kiWEc2IjYkSSJXR0YlSSJuR0YlRiVGJUMlPjglLV9JLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVJKkRpbWVuc2lvbkdGJTYjOSQ+OCQtSSZldmFsZkdGMDYjLCZGNCIiIiomLV9GLkklTm9ybUdGJTYkRjQiIiNGOy1JI0UxR0YlNiNGK0Y7ISIiLCYtX0YuSS9JZGVudGl0eU1hdHJpeEdGJUZERjstSSIuR0YvNiQsJComLUZLNiQtX0YuSSpUcmFuc3Bvc2VHRiU2I0Y2RjZGRUY2RjtGQUZRRkVGJUYlRiU=

HH(<1,2,3>);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKHdWZiQ=

%.<1,2,3>;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIpQzR5Ng==

La fonction suivante donne la d\303\251composition QR, par la m\303\251thode de Householder. Cet algorithme gagnerait \303\240 \303\252tre optimis\303\251 ... QRHouseholder:=proc(m) local q,r,i,n,s; n:=ColumnDimension(m); r:=m; q:=IdentityMatrix(n); for i from 1 to n-1 do s:=HH(r[i..n,i]); s:=DiagonalMatrix([IdentityMatrix(i-1),s]); q:=s.q; r:=s.r; od; [r,Transpose(q)]; end;

Zio2I0kibUc2IjYnSSJxR0YlSSJyR0YlSSJpR0YlSSJuR0YlSSJzR0YlRiVGJUMnPjgnLV9JLkxpbmVhckFsZ2VicmFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiVJMENvbHVtbkRpbWVuc2lvbkdGJTYjOSQ+OCVGNz44JC1fRjFJL0lkZW50aXR5TWF0cml4R0YlNiNGLj8oOCYiIiJGQiwmRi5GQiEiIkZCSSV0cnVlR0YzQyY+OCgtSSNISEdGJTYjJkY5NiQ7RkFGLkZBPkZILV9GMUkvRGlhZ29uYWxNYXRyaXhHRiU2IzckLUY9NiMsJkZBRkJGREZCRkg+RjstSSIuR0YyNiRGSEY7PkY5LUZaNiRGSEY5NyRGOS1fRjFJKlRyYW5zcG9zZUdGJTYjRjtGJUYlRiU=

QR:=QRHouseholder(m);

NyQtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCIoc0BNIy1GJDYjL0YrIilzVD82

%[2].%[1];

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSllRzIi

m;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKXNUXTU=

Transpose(QR[2]).QR[2];

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKVNkKj4i

QRDecomposition(m);

NiQtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCIob0srIi1GJDYjL0YrIiclW2oi

%[1].%[2];

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKTM5bjY=

TTdSMApJNVJUQUJMRV9TQVZFLzEwNTA0MTcyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiIiIiMiIiRGJyIiIUYnRipGKEYnRiY=TTdSMApJNVJUQUJMRV9TQVZFLzExMTYwMzgwWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQkIisoUWQ7dSQhIioiIiFGKiQiK29cL3A1RikkIispKjQ/ZSMqISM1RiokIiskcEczKD1GKSQhK15NNyEzIkYpJCIrIXAtTngmRi9GJg==TTdSMApJNFJUQUJMRV9TQVZFLzI2MjI4NjhYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCQiMywrKys+Q2hzRSEjPSQiMy4rKytRW0FYYEYpJCIzMCsrK2RzJHksKUYpJCIzVSI9dSEqXG5ecihGKSQhM0hUOyY9K01APCdGKSQiM2VSREEoXExJYSJGKSQiM18jZlNOci1OeCZGKSQiM1NCTXF0RV10ZEYpJCEzYEkpZnZvLU54JkYpRiY=TTdSMApJNVJUQUJMRV9TQVZFLzExMzQ5NzU2WCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjKiIkIiQzRkYwMDAwMDAwMDBGNTBBNDAwMDAwMDAwMDAwRjUwQTQwMDgwMDAwMDAwMTZGOTAzRkYwMDAwMDAwMEQyNTE4QkRFNzE5OTgwMDAwMDAwMDNGRjAwMDAwMDAwMTk4NEJCREQyOEM5NjAwMDAwMDAwM0ZGRkZGRkZGRkVEOEU0MzNGRjAwMDAwMDAwMkZGRURGJg==TTdSMApJNVJUQUJMRV9TQVZFLzEwNjY2OTQwWCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjKiIkIiQzRkVGRkZGRkZGRjMzOEI2QkRGN0NFMUNFMDAwMDAwMEJEQzk3MkUwMDAwMDAwMDBCREY3Q0UxQ0UwMDAwMDAwM0ZGMDAwMDAwMDIxQzNENzNFMjg5MDg5NUMwMDAwMDBCREM5NzJFMDAwMDAwMDAwM0UyODkwODk1QzAwMDAwMDNGRUZGRkZGRkZGMUY0QzZGJg==TTdSMApJNFJUQUJMRV9TQVZFLzIyNDk3MjhYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCIiIiIiIUYoRihGJ0YoRihGKEYnRiY=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TTdSMApJNVJUQUJMRV9TQVZFLzEwNzI4NTg4WCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjKiIkIiQzRkYwMDAwMDAwMUM4MEUxM0ZGRkZGRkZGRkVCMzUwNzQwMDdGRkZGRkZGMDY3QzQzRkYwMDAwMDAwMDEzNzU1QkRDOEQ1NTlCREQxMjRGMzNGRjAwMDAwMDAwMEM0Q0YzREYyOTMwNDQwMDAwMDAwM0ZGRkZGRkZGRkY2NTg5QjNGRUZGRkZGRkZDRkY4RUNGJg==TTdSMApJNVJUQUJMRV9TQVZFLzEwNTA0MTcyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiIiIiMiIiRGJyIiIUYnRipGKEYnRiY=TTdSMApJNVJUQUJMRV9TQVZFLzExOTk1NzQwWCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjKiIkIiQzRkVGRkZGRkZGRjBDM0E3M0RFMDBBQzJDMDAwMDAwMDNERDgwMjcxMDAwMDAwMDAzREUwMEFDMkMwMDAwMDAwM0ZFRkZGRkZGRkYzRTVGN0JERTEyQ0YwNjAwMDAwMDAzREQ4MDI3MTAwMDAwMDAwQkRFMTJDRjA2MDAwMDAwMDNGRUZGRkZGRkZGQUQ2MEFGJg==TTdSMApJNFJUQUJMRV9TQVZFLzEwMDMyNjhYLCUpYW55dGhpbmdHNiI2IltnbCEiJSEhISMqIiQiJCwkKiQiIzkjIiIiIiIjI0YrRiksJEYoI0YrIiIoLCRGKCMiIiRGKSwkKiQiI1VGKiMiIiZGNiwkRjUjISIjIiNALCRGNSNGK0Y2LCQqJEYzRiojRitGM0Y/LCRGQCMhIiJGM0YmTTdSMApJM1JUQUJMRV9TQVZFLzE2MzQ4NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyoiJCIkKiQiIzkjIiIiIiIjIiIhRiwsJEYnI0YrIiIoLCQqJCIjVUYpI0YqRi9GLCwkRidGKSwkRjEjISIiIiInLCQqJCIiJEYpI0YqRjtGJg==TTdSMApJNVJUQUJMRV9TQVZFLzExNjcxNDA4WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQiIiIiIiMiIiRGJyIiIUYnRipGKEYnRiY=