︠c4f5d389-694d-4f05-9c5b-9d474b1e048d︠ y=3/5 ︡6d301a73-f5ae-4ba0-b278-718d4ca0136c︡︡{"done":true} ︠c5ac5e42-9dad-480d-b539-48c44e21c89f︠ y.n() ︡1944c218-b754-4670-9fcc-5e7ab905f0e6︡︡{"stdout":"0.600000000000000\n","done":false}︡{"done":true} ︠0baea8cf-28ce-4626-af3b-1bd3a29aafbe︠ type(y) ︡bd488588-a578-4060-8258-c369077acf8b︡︡{"stdout":"\n","done":false}︡{"done":true} ︠f0a20ad2-c472-4bbf-89f6-132d9dc75d2a︠ load("test_integers.py") ︡08132d36-bcc0-4380-bad2-095f22b44bcc︡︡{"stdout":"3/5 = 0\n","done":false}︡{"done":true} ︠b6846e43-5ead-48d0-b65b-bbadd137c012︠ x ︡1a0a88bd-d803-4d65-96b4-a7a354a22c91︡︡{"stdout":"0\n","done":false}︡{"done":true} ︠9b10c30a-ef43-481d-bf7e-55005c8a4312︠ A = [[0 for j in range(15180)] for i in range(26620)] ︡09e71bc0-5882-46a2-acda-b878bb7b2458︡ ︠80515978-6951-4425-8551-c8af8f03ea48︠ #I am interpolating the uniformly space points, chevyshev and legendre polynomial over the interval [-1,2] with the lagrange form then finding the l^2 and l^inf norms to find the error. I am not sure what is going wrong.... OP0=-1 OP1=0 OP2=1 OP3=2 ONUME0=(x-OP1)*(x-OP2)*(x-OP3) ONUME1=(x-OP0)*(x-OP2)*(x-OP3) ONUME2=(x-OP0)*(x-OP1)*(x-OP3) ONUME3=(x-OP0)*(x-OP1)*(x-OP3) ODENO0=(OP0-OP1)*(OP0-OP2)*(OP0-OP3) ODENO1=(OP1-OP0)*(OP1-OP2)*(OP1-OP3) ODENO2=(OP2-OP0)*(OP2-OP1)*(OP2-OP3) ODENO3=(OP3-OP0)*(OP3-OP1)*(OP3-OP2) a=-1 b=2 n=3 i0=0 i1=1 i2=2 i3=3 A=((a+b)/2) B=((b-a)/2) E=(e^(-x)) OCLagr0=(ONUME0/ODENO0) OCLagr1=B*(ONUME1/ODENO1) OCLagr2=B*(ONUME2/ODENO2) OCLagr3=B*(ONUME3/ODENO3) Op0=(e^(-OCLagr0)) Op1=(e^(-OCLagr1)) Op2=(e^(-OCLagr2)) Op3=(e^(-OCLagr3)) Uniformly=((Op0*OCLagr0)+(OP1*OCLagr1)+(Op2*OCLagr2)+(Op3*OCLagr3)) #Lagrange Cheveyshev C0= numerical_approx(cos((pi)*((2*i0+1)/(2*n+2)))) C1= numerical_approx(cos((pi)*((2*i1+1)/(2*n+2)))) C2= numerical_approx(cos((pi)*((2*i2+1)/(2*n+2)))) C3= numerical_approx(cos((pi)*((2*i3+1)/(2*n+2)))) CV0=((B*C0)+A) CV1=((B*C1)+A) CV2=((B*C2)+A) CV3=((B*C3)+A) xo=CV0 xi=CV1 xii=CV2 xiii=CV3 NUME0=(x-xi)*(x-xii)*(x-xiii) NUME1=(x-xo)*(x-xii)*(x-xiii) NUME2=(x-xo)*(x-xi)*(x-xiii) NUME3=(x-xo)*(x-xi)*(x-xii) DENO0=(xo-xi)*(xo-xii)*(xo-xiii) DENO1=(xi-xo)*(xi-xii)*(xi-xiii) DENO2=(xii-xo)*(xii-xi)*(xii-xiii) DENO3=(xiii-xo)*(xiii-xi)*(xiii-xii) CLagr0=(NUME0/DENO0) CLagr1=(NUME1/DENO1) CLagr2=(NUME2/DENO2) CLagr3=(NUME3/DENO3) P0=numerical_approx(e^(-(C0))) P1=numerical_approx(e^(-(C1))) P2=numerical_approx(e^(-(C2))) P3=numerical_approx(e^(-(C3))) F0=P0*CLagr0 F1=P1*CLagr1 F2=P2*CLagr2 F3=P3*CLagr3 Chevyshev=F0+F1+F2+F3 ##Legendre LP0=1 LP1=x LP2=(1/2)*(3*((x)^2)-1) LP3=(((1/2)*(((5)*x^2)-((3)*x)))) LP4=((1/8)*(((35)*x^4)-((30)*x^2)+3)) f=7*LP0 g=8*LP4 h=20*LP2 j=3*LP1 k=2*LP3 X1=numerical_approx(-sqrt(2/35*sqrt(30) + 3/7)) X2=numerical_approx(sqrt(2/35*sqrt(30) + 3/7)) X3=numerical_approx(-sqrt(-2/35*sqrt(30) + 3/7)) X4=numerical_approx(sqrt(-2/35*sqrt(30) + 3/7)) LOV0=((B*X1)+A) LOV1=((B*X2)+A) LOV2=((B*X3)+A) LOV3=((B*X4)+A) lxo=LOV0 lxi=LOV1 lxii=LOV2 lxiii=LOV3 LNUME0=(x-lxi)*(x-lxii)*(x-lxiii) LNUME1=(x-lxo)*(x-lxii)*(x-lxiii) LNUME2=(x-lxo)*(x-lxi)*(x-lxiii) LNUME3=(x-lxo)*(x-lxi)*(x-lxii) LDENO0=(lxo-lxi)*(lxo-lxii)*(lxo-lxiii) LDENO1=(lxi-lxo)*(lxi-lxii)*(lxi-lxiii) LDENO2=(lxii-lxo)*(lxii-lxi)*(lxii-lxiii) LDENO3=(lxiii-lxo)*(lxiii-lxi)*(lxiii-lxii) LLagr0=(LNUME0/LDENO0) LLagr1=(LNUME1/LDENO1) LLagr2=(LNUME2/LDENO2) LLagr3=(LNUME3/LDENO3) LP0=numerical_approx(e^(-(X1))) LP1=numerical_approx(e^(-(X2))) LP2=numerical_approx(e^(-(X3))) LP3=numerical_approx(e^(-(X4))) LF0=LP0*LLagr0 LF1=LP1*LLagr1 LF2=LP2*LLagr2 LF3=LP3*LLagr3 Legendere=LF0+LF1+LF2+LF3 f = (0.116167595984182*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.00997156537728) - 0.495492470126893*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.79170446739108) + 0.978004473650812*(x + 0.791704467391079)*(x - 1.00997156537728)*(x - 1.79170446739108) - 0.650217267384311*(x + 0.00997156537728427)*(x - 1.00997156537728)*(x - 1.79170446739108))^2 g=(e^-x)^2 h=(0.0900243711092997*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.07402514854763) - 0.373399955130276*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.88581929876693) + 0.802728841958563*(x + 0.885819298766930)*(x - 1.07402514854763)*(x - 1.88581929876693) - 0.571257167284918*(x + 0.0740251485476346)*(x - 1.07402514854763)*(x - 1.88581929876693))^2 u=-3/4*(x + 1)*(x - 2)*x*e^(3/4*(x + 1)*(x - 2)*x) + 1/4*(x + 1)*(x - 2)*x*e^(-1/4*(x + 1)*(x - 2)*x) - 1/6*(x - 1)*(x - 2)*x*e^(1/6*(x - 1)*(x - 2)*x) asw=sum(f,x,(-1/1),(2/1))#Chevy ert=numerical_approx(sum(g,x,-1,2/1))#e^-x cvb=sum(h,x,-1/1,2/1)#Leg plm=sum(u,x,-1,2/1)#Uni l2C=norm(vector([ert-asw])) l2L=norm(vector([ert-cvb])) l2O=norm(vector([ert-plm])) l2E=norm(vector([ert])) linfC=abs(max([asw])) linfL=abs(max([cvb])) linfO=abs(max([plm])) linfE=abs(max([ert])) l2C l2L l2O linfC linfL linfO ︡dffd4511-7ca6-41a5-ad67-797fb1cc3539︡︡{"stdout":"1.37065555196861\n","done":false}︡{"stdout":"1.39852837567751\n","done":false}︡{"stdout":"sqrt((1/2*(e^2 - 2*e^(1/2) - 3)*e^(-3/2) + 8.54270702105600)^2)\n","done":false}︡{"stdout":"9.913362573024607\n","done":false}︡{"stdout":"9.941235396733509\n","done":false}︡{"stdout":"1/2*abs(e^2 - 2*e^(1/2) - 3)*e^(-3/2)\n","done":false}︡{"done":true} ︠771987a5-8154-4ac6-b8da-0158700c9894s︠ var("x") OP0=-1 OP1=0 OP2=1 OP3=2 ONUME0=(x-OP1)*(x-OP2)*(x-OP3) ONUME1=(x-OP0)*(x-OP2)*(x-OP3) ONUME2=(x-OP0)*(x-OP1)*(x-OP3) ONUME3=(x-OP0)*(x-OP1)*(x-OP3) ODENO0=(OP0-OP1)*(OP0-OP2)*(OP0-OP3) ODENO1=(OP1-OP0)*(OP1-OP2)*(OP1-OP3) ODENO2=(OP2-OP0)*(OP2-OP1)*(OP2-OP3) ODENO3=(OP3-OP0)*(OP3-OP1)*(OP3-OP2) OCLagr0=(ONUME0/ODENO0) OCLagr1=(ONUME1/ODENO1) OCLagr2=(ONUME2/ODENO2) OCLagr3=(ONUME3/ODENO3) Op0=(e^(-OP0)) Op1=(e^(-OP1)) Op2=(e^(-OP2)) Op3=(e^(-OP3)) Uniformly=((Op0*OCLagr0)+(OP1*OCLagr1)+(Op2*OCLagr2)+(Op3*OCLagr3)) ︡ec0e7820-b4dc-4a10-b928-243fff45910f︡︡{"stdout":"x\n","done":false}︡{"done":true} ︠57c63823-6488-46ac-adca-be6b6d92138fs︠ Uniformly ︡5f309af5-136c-491e-849b-8339cd7cbd6c︡︡{"stdout":"-1/6*(x - 1)*(x - 2)*x*e - 1/2*(x + 1)*(x - 2)*x*e^(-1) + 1/6*(x + 1)*(x - 2)*x*e^(-2)\n","done":false}︡{"done":true} ︠8beef2a7-ef0c-48b2-bd21-121e321ed868s︠ plot(Uniformly,(-1,2))+plot(e^(-x),-1,2,color="red") ︡5c923527-b86e-428f-8261-e31efa0a72a7︡︡{"once":false,"done":false,"file":{"show":true,"uuid":"f1917e48-54f5-488b-af96-7f42d9fe9e58","filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute2-us/7270/tmp_s4gy75.svg"}}︡{"html":"
","done":false}︡{"done":true} ︠eceb1952-9daf-4765-95df-4eb7752ae26cs︠ OP0=-1 OP1=0 OP2=1 OP3=2 ONUME0=(x-OP1)*(x-OP2)*(x-OP3) ONUME1=(x-OP0)*(x-OP2)*(x-OP3) ONUME2=(x-OP0)*(x-OP1)*(x-OP3) ONUME3=(x-OP0)*(x-OP1)*(x-OP3) ODENO0=(OP0-OP1)*(OP0-OP2)*(OP0-OP3) ODENO1=(OP1-OP0)*(OP1-OP2)*(OP1-OP3) ODENO2=(OP2-OP0)*(OP2-OP1)*(OP2-OP3) ODENO3=(OP3-OP0)*(OP3-OP1)*(OP3-OP2) OCLagr0=(ONUME0/ODENO0) OCLagr1=(ONUME1/ODENO1) OCLagr2=(ONUME2/ODENO2) OCLagr3=(ONUME3/ODENO3) Op0=(e^(-OP0)) Op1=(e^(-OP1)) Op2=(e^(-OP2)) Op3=(e^(-OP3)) Uniformly=((Op0*OCLagr0)+(OP1*OCLagr1)+(Op2*OCLagr2)+(Op3*OCLagr3)) #Lagrange Cheveyshev a=-1 b=2 n=3 i0=0 i1=1 i2=2 i3=3 A=((a+b)/2) B=((b-a)/2) E=(e^(-x)) C0= numerical_approx(cos((pi)*((2*i0+1)/(2*n+2)))) C1= numerical_approx(cos((pi)*((2*i1+1)/(2*n+2)))) C2= numerical_approx(cos((pi)*((2*i2+1)/(2*n+2)))) C3= numerical_approx(cos((pi)*((2*i3+1)/(2*n+2)))) CV0=((B*C0)+A) CV1=((B*C1)+A) CV2=((B*C2)+A) CV3=((B*C3)+A) xo=CV0 xi=CV1 xii=CV2 xiii=CV3 NUME0=(x-xi)*(x-xii)*(x-xiii) NUME1=(x-xo)*(x-xii)*(x-xiii) NUME2=(x-xo)*(x-xi)*(x-xiii) NUME3=(x-xo)*(x-xi)*(x-xii) DENO0=(xo-xi)*(xo-xii)*(xo-xiii) DENO1=(xi-xo)*(xi-xii)*(xi-xiii) DENO2=(xii-xo)*(xii-xi)*(xii-xiii) DENO3=(xiii-xo)*(xiii-xi)*(xiii-xii) CLagr0=(NUME0/DENO0) CLagr1=(NUME1/DENO1) CLagr2=(NUME2/DENO2) CLagr3=(NUME3/DENO3) P0=numerical_approx(e^(-(C0))) P1=numerical_approx(e^(-(C1))) P2=numerical_approx(e^(-(C2))) P3=numerical_approx(e^(-(C3))) F0=P0*CLagr0 F1=P1*CLagr1 F2=P2*CLagr2 F3=P3*CLagr3 Chevyshev=F0+F1+F2+F3 ##Legendre LP0=1 LP1=x LP2=(1/2)*(3*((x)^2)-1) LP3=(((1/2)*(((5)*x^2)-((3)*x)))) LP4=((1/8)*(((35)*x^4)-((30)*x^2)+3)) f=7*LP0 g=8*LP4 h=20*LP2 j=3*LP1 k=2*LP3 solve(LP4,x) X1=numerical_approx(-sqrt(2/35*sqrt(30) + 3/7)) X2=numerical_approx(sqrt(2/35*sqrt(30) + 3/7)) X3=numerical_approx(-sqrt(-2/35*sqrt(30) + 3/7)) X4=numerical_approx(sqrt(-2/35*sqrt(30) + 3/7)) LOV0=((B*X1)+A) LOV1=((B*X2)+A) LOV2=((B*X3)+A) LOV3=((B*X4)+A) lxo=LOV0 lxi=LOV1 lxii=LOV2 lxiii=LOV3 LNUME0=(x-lxi)*(x-lxii)*(x-lxiii) LNUME1=(x-lxo)*(x-lxii)*(x-lxiii) LNUME2=(x-lxo)*(x-lxi)*(x-lxiii) LNUME3=(x-lxo)*(x-lxi)*(x-lxii) LDENO0=(lxo-lxi)*(lxo-lxii)*(lxo-lxiii) LDENO1=(lxi-lxo)*(lxi-lxii)*(lxi-lxiii) LDENO2=(lxii-lxo)*(lxii-lxi)*(lxii-lxiii) LDENO3=(lxiii-lxo)*(lxiii-lxi)*(lxiii-lxii) LLagr0=(LNUME0/LDENO0) LLagr1=(LNUME1/LDENO1) LLagr2=(LNUME2/LDENO2) LLagr3=(LNUME3/LDENO3) LP0=numerical_approx(e^(-(X1))) LP1=numerical_approx(e^(-(X2))) LP2=numerical_approx(e^(-(X3))) LP3=numerical_approx(e^(-(X4))) LF0=LP0*LLagr0 LF1=LP1*LLagr1 LF2=LP2*LLagr2 LF3=LP3*LLagr3 Legendere=LF0+LF1+LF2+LF3 f = (0.116167595984182*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.00997156537728) - 0.495492470126893*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.79170446739108) + 0.978004473650812*(x + 0.791704467391079)*(x - 1.00997156537728)*(x - 1.79170446739108) - 0.650217267384311*(x + 0.00997156537728427)*(x - 1.00997156537728)*(x - 1.79170446739108))^2 g=(e^-x)^2 h=(0.0900243711092997*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.07402514854763) - 0.373399955130276*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.88581929876693) + 0.802728841958563*(x + 0.885819298766930)*(x - 1.07402514854763)*(x - 1.88581929876693) - 0.571257167284918*(x + 0.0740251485476346)*(x - 1.07402514854763)*(x - 1.88581929876693))^2 u=-3/4*(x + 1)*(x - 2)*x*e^(3/4*(x + 1)*(x - 2)*x) + 1/4*(x + 1)*(x - 2)*x*e^(-1/4*(x + 1)*(x - 2)*x) - 1/6*(x - 1)*(x - 2)*x*e^(1/6*(x - 1)*(x - 2)*x) asw=sum(f,x,(-1/1),(2/1))#Chevy ert=numerical_approx(sum(g,x,-1,2/1))#e^-x cvb=sum(h,x,-1/1,2/1)#Leg plm=sum(u,x,-1,2/1)#Uni l2C=norm(vector([ert-asw])) l2L=norm(vector([ert-cvb])) l2O=norm(vector([ert-plm])) l2E=norm(vector([ert])) linfC=abs(max([asw])) linfL=abs(max([cvb])) linfO=abs(max([plm])) linfE=abs(max([ert])) l2C l2L l2O linfC linfL linfO Chevyshev Uniformly Legendere plot(Legendere) l2L linfL print'The first root is %s'%X1 print'The second root is %s'%X2 print'The third root is %s'%X3 print'The forth root is %s'%X4 Chevyshev plot(Chevyshev) l2C linfC print'The first Chevyshev point is %s'%CV0 print'The second Chevyshev point is %s'%CV1 print'The third Chevyshev point is %s'%CV2 print'The forth Chevyshev point is %s'%CV3 Uniformly plot(Uniformly) l2O linfO ︡ece336f7-64de-4f92-8aab-2e541aeb54ab︡︡{"stdout":"[x == -sqrt(2/35*sqrt(30) + 3/7), x == sqrt(2/35*sqrt(30) + 3/7), x == -sqrt(-2/35*sqrt(30) + 3/7), x == sqrt(-2/35*sqrt(30) + 3/7)]","done":false}︡{"stdout":"\n","done":false}︡{"stdout":"1.37065555196861\n","done":false}︡{"stdout":"1.39852837567751\n","done":false}︡{"stdout":"sqrt((1/2*(e^2 - 2*e^(1/2) - 3)*e^(-3/2) + 8.54270702105600)^2)\n","done":false}︡{"stdout":"9.913362573024607\n","done":false}︡{"stdout":"9.941235396733509\n","done":false}︡{"stdout":"1/2*abs(e^2 - 2*e^(1/2) - 3)*e^(-3/2)\n","done":false}︡{"stdout":"0.0900243711092997*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.07402514854763) - 0.373399955130276*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.88581929876693) + 0.802728841958563*(x + 0.885819298766930)*(x - 1.07402514854763)*(x - 1.88581929876693) - 0.571257167284918*(x + 0.0740251485476346)*(x - 1.07402514854763)*(x - 1.88581929876693)\n","done":false}︡{"stdout":"-1/6*(x - 1)*(x - 2)*x*e - 1/2*(x + 1)*(x - 2)*x*e^(-1) + 1/6*(x + 1)*(x - 2)*x*e^(-2)\n","done":false}︡{"stdout":"0.116167595984182*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.00997156537728) - 0.495492470126893*(x + 0.791704467391079)*(x + 0.00997156537728427)*(x - 1.79170446739108) + 0.978004473650812*(x + 0.791704467391079)*(x - 1.00997156537728)*(x - 1.79170446739108) - 0.650217267384311*(x + 0.00997156537728427)*(x - 1.00997156537728)*(x - 1.79170446739108)\n","done":false}︡{"once":false,"done":false,"file":{"show":true,"uuid":"caae6524-fec7-48ee-a689-4103cd51b425","filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute2-us/7270/tmp_4qPMJ2.svg"}}︡{"html":"
","done":false}︡{"stdout":"1.39852837567751\n","done":false}︡{"stdout":"9.941235396733509\n","done":false}︡{"stdout":"The first root is -0.861136311594053\n","done":false}︡{"stdout":"The second root is 0.861136311594053\n","done":false}︡{"stdout":"The third root is -0.339981043584856\n","done":false}︡{"stdout":"The forth root is 0.339981043584856\n","done":false}︡{"stdout":"0.0900243711092997*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.07402514854763) - 0.373399955130276*(x + 0.885819298766930)*(x + 0.0740251485476346)*(x - 1.88581929876693) + 0.802728841958563*(x + 0.885819298766930)*(x - 1.07402514854763)*(x - 1.88581929876693) - 0.571257167284918*(x + 0.0740251485476346)*(x - 1.07402514854763)*(x - 1.88581929876693)\n","done":false}︡{"once":false,"done":false,"file":{"show":true,"uuid":"4aa8c9fd-5d87-48c1-98ae-6934cd8a96e7","filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute2-us/7270/tmp_oXFysc.svg"}}︡{"html":"
","done":false}︡{"stdout":"1.37065555196861\n","done":false}︡{"stdout":"9.913362573024607\n","done":false}︡{"stdout":"The first Chevyshev point is 1.88581929876693\n","done":false}︡{"stdout":"The second Chevyshev point is 1.07402514854763\n","done":false}︡{"stdout":"The third Chevyshev point is -0.0740251485476346\n","done":false}︡{"stdout":"The forth Chevyshev point is -0.885819298766930\n","done":false}︡{"stdout":"-1/6*(x - 1)*(x - 2)*x*e - 1/2*(x + 1)*(x - 2)*x*e^(-1) + 1/6*(x + 1)*(x - 2)*x*e^(-2)\n","done":false}︡{"once":false,"done":false,"file":{"show":true,"uuid":"f6ce1cea-b271-4a68-886b-0bc28e0e1433","filename":"/projects/4599068b-1689-4a77-81c7-9cd760e0f06d/.sage/temp/compute2-us/7270/tmp_X5540b.svg"}}︡{"html":"
","done":false}︡{"stdout":"sqrt((1/2*(e^2 - 2*e^(1/2) - 3)*e^(-3/2) + 8.54270702105600)^2)\n","done":false}︡{"stdout":"1/2*abs(e^2 - 2*e^(1/2) - 3)*e^(-3/2)\n","done":false}︡{"done":true} ︠a027d8b8-320e-4bf7-b441-fcda88eec85as︠ #want 200 equally spaced points from -1 to 2 delta=(2-(-1))/200 ︡f864dae5-b0f6-47a5-98be-3fb78b10c336︡︡{"done":true} ︠97f4eadc-ba61-4126-bba0-0c9740a878a3s︠ points = [-1+i*delta.n(digits=3) for i in range(0,200)] ︡1c43455c-8d17-4304-8514-739f42e11b78︡︡{"done":true} ︠d9c92236-99c1-4396-a525-978b06ba7a47s︠ points ︡e049b126-6219-4d4f-8316-90a99d0dbbf2︡︡{"stdout":"[-1.00, -0.985, -0.970, -0.955, -0.940, -0.925, -0.910, -0.895, -0.880, -0.865, -0.850, -0.835, -0.820, -0.805, -0.790, -0.775, -0.760, -0.745, -0.730, -0.715, -0.700, -0.685, -0.670, -0.655, -0.640, -0.625, -0.610, -0.595, -0.580, -0.565, -0.550, -0.535, -0.520, -0.505, -0.490, -0.475, -0.460, -0.445, -0.430, -0.415, -0.400, -0.385, -0.370, -0.355, -0.340, -0.325, -0.310, -0.295, -0.280, -0.265, -0.250, -0.235, -0.220, -0.205, -0.190, -0.175, -0.160, -0.145, -0.130, -0.115, -0.100, -0.0850, -0.0700, -0.0550, -0.0400, -0.0250, -0.00995, 0.00500, 0.0200, 0.0350, 0.0500, 0.0651, 0.0801, 0.0950, 0.110, 0.125, 0.140, 0.155, 0.170, 0.185, 0.200, 0.215, 0.230, 0.245, 0.260, 0.275, 0.290, 0.305, 0.320, 0.335, 0.350, 0.365, 0.380, 0.395, 0.410, 0.425, 0.440, 0.455, 0.470, 0.485, 0.500, 0.515, 0.530, 0.545, 0.560, 0.575, 0.590, 0.605, 0.620, 0.635, 0.650, 0.665, 0.680, 0.695, 0.710, 0.725, 0.740, 0.755, 0.770, 0.785, 0.800, 0.815, 0.830, 0.845, 0.860, 0.875, 0.890, 0.905, 0.920, 0.935, 0.950, 0.965, 0.980, 0.995, 1.01, 1.03, 1.04, 1.05, 1.07, 1.08, 1.10, 1.11, 1.13, 1.15, 1.16, 1.18, 1.19, 1.21, 1.22, 1.24, 1.25, 1.27, 1.28, 1.30, 1.31, 1.32, 1.34, 1.35, 1.37, 1.39, 1.40, 1.42, 1.43, 1.45, 1.46, 1.48, 1.49, 1.51, 1.52, 1.54, 1.55, 1.56, 1.58, 1.59, 1.61, 1.62, 1.64, 1.66, 1.67, 1.69, 1.70, 1.72, 1.73, 1.75, 1.76, 1.78, 1.79, 1.81, 1.82, 1.83, 1.85, 1.86, 1.88, 1.90, 1.91, 1.93, 1.94, 1.96, 1.97, 1.99]\n","done":false}︡{"done":true} ︠16b2864b-8f8d-4f1c-a985-fb4327f5992es︠ type(points[1]) ︡a67a8770-2240-4a87-a523-f27e018f2512︡︡{"stdout":"\n","done":false}︡{"done":true} ︠f9e4578b-e519-44c6-9ad1-00a571680722s︠ pp=points[1] ︡a8234b1c-8fe9-4835-a15a-9a575b7254ac︡︡{"done":true} ︠f85af3f5-3b7b-4e9f-b268-4aa8bb880a7cs︠ pp.n(prec=10) ︡114addec-14b5-40da-a86d-3c84a2de43bb︡︡{"stdout":"-0.99\n","done":false}︡{"done":true} ︠ac5aa0a1-4ebc-40a4-b3ca-e5670da2bcc8s︠ delta.n() abs ︡75eb5e4d-6f8e-407f-aaf4-4229282aa51e︡︡{"stdout":"0.0150000000000000\n","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠cf0ba0d3-907a-4785-ac76-731200a17e84s︠ L=[3,4,5] L2norm_of_l=sqrt(sum([x^2 for x in L])) ︡7df813d0-730d-4b6a-803f-63a766628d7e︡︡{"done":true} ︠d549cbf1-66e6-4978-949e-6a47e488d0c6s︠ max(L) ︡d065e2f9-7e88-4304-9c61-edee1a08fc70︡︡{"stdout":"5\n","done":false}︡{"done":true} ︠3cb86418-3591-4f5e-a79b-ac68272d6b7ds︠ L2norm_of_l ︡570153d1-078d-4501-a865-13f47ea0d15d︡︡{"stdout":"5*sqrt(2)\n","done":false}︡{"done":true} ︠5fa0365a-ce6b-46b5-9648-742061de86e4s︠ norm? ︡92eef71b-d5d9-4ac0-a557-964c0e7abc8c︡︡{"code":{"source":"File: /projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/misc/functional.py\nSignature : norm()\nDocstring :\nReturns the norm of \"x\".\n\nFor matrices and vectors, this returns the L2-norm. The L2-norm of\na vector *v* = (v_1, v_2, ..., v_n), also called the Euclidean\nnorm, is defined as\n\n |*v*| = sqrt{sum_{i=1}^n |v_i|^2}\n\nwhere |v_i| is the complex modulus of v_i. The Euclidean norm is\noften used for determining the distance between two points in two-\nor three-dimensional space.\n\nFor complex numbers, the function returns the field norm. If c = a\n+ bi is a complex number, then the norm of c is defined as the\nproduct of c and its complex conjugate\n\n norm(c) = norm(a + bi) = c * overline{c} = a^2 + b^2.\n\nThe norm of a complex number is different from its absolute value.\nThe absolute value of a complex number is defined to be the square\nroot of its norm. A typical use of the complex norm is in the\nintegral domain ZZ[i] of Gaussian integers, where the norm of each\nGaussian integer c = a + bi is defined as its complex norm.\n\nSee also: * \"sage.matrix.matrix2.Matrix.norm()\"\n\n\n * \"sage.modules.free_module_element.FreeModuleElement.norm()\"\n\n * \"sage.rings.complex_double.ComplexDoubleElement.norm()\"\n\n * \"sage.rings.complex_number.ComplexNumber.norm()\"\n\n * \"sage.symbolic.expression.Expression.norm()\"\n\nEXAMPLES:\n\nThe norm of vectors:\n\n sage: z = 1 + 2*I\n sage: norm(vector([z]))\n sqrt(5)\n sage: v = vector([-1,2,3])\n sage: norm(v)\n sqrt(14)\n sage: _ = var(\"a b c d\", domain='real')\n sage: v = vector([a, b, c, d])\n sage: norm(v)\n sqrt(a^2 + b^2 + c^2 + d^2)\n\nThe norm of matrices:\n\n sage: z = 1 + 2*I\n sage: norm(matrix([[z]]))\n 2.23606797749979\n sage: M = matrix(ZZ, [[1,2,4,3], [-1,0,3,-10]])\n sage: norm(M) # abs tol 1e-14\n 10.690331129154467\n sage: norm(CDF(z))\n 5.0\n sage: norm(CC(z))\n 5.00000000000000\n\nThe norm of complex numbers:\n\n sage: z = 2 - 3*I\n sage: norm(z)\n 13\n sage: a = randint(-10^10, 100^10)\n sage: b = randint(-10^10, 100^10)\n sage: z = a + b*I\n sage: bool(norm(z) == a^2 + b^2)\n True\n\nThe complex norm of symbolic expressions:\n\n sage: a, b, c = var(\"a, b, c\")\n sage: assume((a, 'real'), (b, 'real'), (c, 'real'))\n sage: z = a + b*I\n sage: bool(norm(z).simplify() == a^2 + b^2)\n True\n sage: norm(a + b).simplify()\n a^2 + 2*a*b + b^2\n sage: v = vector([a, b, c])\n sage: bool(norm(v).simplify() == sqrt(a^2 + b^2 + c^2))\n True\n sage: forget()","mode":"text/x-rst","lineno":-1,"filename":null},"done":false}︡{"done":true} ︠cb675a8d-4008-4dab-9621-d91a4b823a97s︠ solns=solve(x^2==1,x) ︡353ccceb-fccc-4ad9-81b0-c7cb4e8b878a︡︡{"done":true} ︠7cb61f72-303b-49dd-935f-fd27ff8dbedes︠ type(solns) ︡6e920868-c167-4254-8792-87b0ff36ccdc︡︡{"stdout":"\n","done":false}︡{"done":true} ︠7473638c-1e8a-41bf-af9d-8bfbd58a26bas︠ solns ︡07cd2566-422d-49c9-8105-f45cd39b0a85︡︡{"stdout":"[x == -1, x == 1]\n","done":false}︡{"done":true} ︠c7557a51-2c2b-43cf-aa3d-77356e4a4050s︠ list(solns) ︡73f4495e-2a07-4548-944a-ed5a09fc0d57︡︡{"stdout":"[x == -1, x == 1]\n","done":false}︡{"done":true} ︠07e2c9e4-531e-4c97-9363-c40d95b0eb34s︠ solns[0] ︡5fd4801d-b974-4e9d-bd49-80ff31fe5d08︡︡{"stdout":"x == -1\n","done":false}︡{"done":true} ︠3cb941ec-5873-4ea4-aaf7-a021b285587as︠ type(solns[0]) ︡a3989bb5-26fe-496f-a7f4-ebbfb43b3be2︡︡{"stdout":"\n","done":false}︡{"done":true} ︠8e42983b-47d3-4d69-ba8a-e2b6cb22b9bas︠ solns[0].rhs() ︡435059f3-fe85-4354-b819-2e602b2d5874︡︡{"stdout":"-1\n","done":false}︡{"done":true} ︠c1a98e49-5f73-4d5e-82d2-c3cab11ea9aas︠ [solns[x].rhs() for x in range(0,len(solns))] ︡6192e9a7-509e-44d9-ab71-9d67c52d9628︡︡{"stdout":"[-1, 1]\n","done":false}︡{"done":true} ︠f6382b2d-b389-44f9-8d8c-f568b2ce9e30︠