Solving recurrence T(n) = T(n/2) + Θ(1) by substitution - recursion

So I understand how to do it when the recurrence looks something like this:
T(n) = 2T(n/2) + n
In that case I would guess the answer to be O(nlogn) and then use induction to prove it. But for this one the Θ(1) throws me off. How would you do it? If you could produce the steps of induction that would be amazing.
Thank you so much!


try substituting the value recursively
T(n) = T(n/2) + Θ(1)
= (T(n/4) + Θ(1)) + Θ(1) = T(n/4) + Θ(1) + Θ(1) = T(n/4) + 2*Θ(1)
= (T(n/8) + Θ(1)) + 2*Θ(1)= T(n/8) + 3*Θ(1)
= T(n/16) + 4*Θ(1)
= T(n/32) + 5*Θ(1) [ T(n/2^5) + 5*Θ(1) ]
.
.
= T(1) + log2(n)*Θ(1)
= O(log2(n))

Related

How can I simplify the formula in R?

I want to estimate a set formulas in R by systemfit,for each formula,it's mathmatic expression can be written as
https://www.mathcha.io/editor/VQKXXuLvCjWClxjyy4Ie0xBKOC1eN3lgCMMXdgG
In stata ,I can handle it by
gen double `lnap' = `a0' + `a1'*`lnp1' + `a2'*`lnp2' + `a3'*`lnp3' + `a4'*`lnp4' + `a5'*`lnp5' + `a6'*`lnp6'
forvalues i = 1/6 {
forvalues j = 1/6 {
replace `lnap' = `lnpindex' + 0.5*`g`i'`j''*`lnp`i''*`lnp`j''
}
}
gen double `bp' = 0
forvalues i = 1/6 {
replace `bp' = `bofp' + `lnp`i''*`b`i''
}
replace `w1' = (`a1' + `g11'*`lnp1' + `g12'*`lnp2' + `g13'*`lnp3' + `g14'*`lnp4' + `g15'*`lnp5' + `g16'*`lnp6' + ///
`b1'*(`lnm' - `lnap') + `l1'/`bp'*(`lnm' - `lnap')^2 + `z1'*`eta11' + `z2'*`eta21' + `z3'*`eta31' + `z4'*`eta41'+ `z5'*`eta51'
I have read manuel for formula,it seems that the only way is to simplify this equation.I wonder if you can tell me how to write this complex equation to formula in R?
Many thanks!

Recurrence relation with not constant coefficients

I have this recurrence relation
L^2 G[p]= 2(p-1)(2p-1)G[p-1] + ((p-1)(p-2)+a^2) G[p-2], where L and a are parameters.
Does anyone could help me to find the solution? Thanks

This looks like quite a complicated calculation, especially when no start values are given.
To get some more insight, one could use sympy, Python's symbolic math library to print the formulas for small values of p:
from sympy import symbols
def func_G(p):
if p == 0:
return G0
elif p == 1:
return G1
else:
return (2 * (p - 1) * (2 * p - 1) * func_G(p - 1) + ((p - 1) * (p - 2) + a ** 2) * func_G(p - 2)) / L ** 2
a, L, G0, G1 = symbols('a L G0 G1')
for p in range(8):
print(p, ':', func_G(p).simplify())
Prints out:
0 : G0
1 : G1
2 : (G0*a**2 + 6*G1)/L**2
3 : (20*G0*a**2 + G1*L**2*(a**2 + 2) + 120*G1)/L**4
4 : (840*G0*a**2 + 42*G1*L**2*(a**2 + 2) + 5040*G1 + L**2*(a**2 + 6)*(G0*a**2 + 6*G1))/L**6
5 : (60480*G0*a**2 + 3024*G1*L**2*(a**2 + 2) + 362880*G1 + 72*L**2*(a**2 + 6)*(G0*a**2 + 6*G1) + L**2*(a**2 + 12)*(20*G0*a**2 + G1*L**2*(a**2 + 2) + 120*G1))/L**8
6 : (G0*L**4*a**6 + 26*G0*L**4*a**4 + 120*G0*L**4*a**2 + 10960*G0*L**2*a**4 + 90720*G0*L**2*a**2 + 6652800*G0*a**2 + 158*G1*L**4*a**4 + 2620*G1*L**4*a**2 + 5040*G1*L**4 + 398400*G1*L**2*a**2 + 1209600*G1*L**2 + 39916800*G1)/L**10
7 : (248*G0*L**4*a**6 + 7488*G0*L**4*a**4 + 38880*G0*L**4*a**2 + 1770240*G0*L**2*a**4 + 15966720*G0*L**2*a**2 + 1037836800*G0*a**2 + G1*L**6*a**6 + 44*G1*L**6*a**4 + 444*G1*L**6*a**2 + 720*G1*L**6 + 28224*G1*L**4*a**4 + 526080*G1*L**4*a**2 + 1088640*G1*L**4 + 62513280*G1*L**2*a**2 + 199584000*G1*L**2 + 6227020800*G1)/L**12

r glmmLasso : Error in n %*% s (glmm.rmd#64): requires numeric/complex matrix/vector arguments

I am trying to fit a glmmLasso model using this code:
lasso <- glmmLasso(stars ~ ADJRIND + AUC + KAPPA + DICE + ICCORR + JACRD + MUTINF + RNDIND + SURFOVLP + SURFDICE + VOLSMTY + HDRFDST + AVGDIST + MAHLNBS + VARINFO + GCOERR + PROBDST + SNSVTY + SPCFTY + PRCISON + RECALL + FMEASR + ACURCY + FALLOUT + TP + FP + TN + FN + GTVOL + SEGVOL, rnd = ~1|participant, family = acat(), data = mixedModel_df, lambda=10, switch.NR=TRUE, control=list(print.iter=TRUE))
I get the following error msg:
Error in n %*% s (glmm.rmd#64): requires numeric/complex matrix/vector arguments
all the IVs and my DV are doubles, only my random variable is a factor.
Also what should I specify for family?
As I requested here the head of my dataframe:
"","condition","algorithm","patient","participant","stars","ADJRIND","AUC","KAPPA","DICE","ICCORR","JACRD","MUTINF","RNDIND","SURFOVLP","SURFDICE","VOLSMTY","HDRFDST","AVGDIST","MAHLNBS","VARINFO","GCOERR","PROBDST","SNSVTY","SPCFTY","PRCISON","RECALL","FMEASR","ACURCY","FALLOUT","TP","FP","TN","FN","GTVOL","SEGVOL"
"1","rnd","BIIPL-rnd","brats_2013_pat0116_1","ablaze_gull",3,0.756964689377408,0.926221010756581,0.768802263900468,0.776064818441724,0.768315078860883,0.65050064332349,0.135475119060726,0.971161689934318,0.874015774267621,0.887817577382146,0.866059384942721,14.6242696243608,0.938245742350098,0.293565667284625,0.160520618779326,0.0249053133772558,0.334382929069828,0.861652251473578,0.990789770039584,0.744264326384239,0.861652251473578,0.776064818441724,0.985270803169769,0.00921022996041633,669.333333333333,198.222222222222,20804.6666666667,131.444444444444,800.777777777778,867.555555555556
"2","simple","simple","brats_2013_pat0116_1","ablaze_gull",6,0.864182045030138,0.954469433017555,0.870656560504094,0.874123730811541,0.870654729849903,0.779756022847205,0.164077165463804,0.986913573761583,0.974186271807427,0.979900170507277,0.957512050579079,6.97899055058698,0.175777792477891,0.114795641459174,0.0924422799237813,0.0125654163033296,0.147516600257771,0.913391157391641,0.99554770864347,0.839991365397643,0.913391157391641,0.874123730811541,0.99340680392773,0.00445229135652993,749.333333333333,92.4444444444444,20910.4444444444,51.4444444444444,800.777777777778,841.777777777778
"3","zyx","zyx","brats_2013_pat0116_1","ablaze_gull",5,0.870755923933255,0.952679073225429,0.876764151799132,0.879962617937312,0.876762586811393,0.78886242309958,0.165135328034368,0.98790705513673,0.981797036327553,0.980082675921329,0.964137575847861,7.45373777515636,0.172965178585364,0.102381275486303,0.0869596431464502,0.0116484532775206,0.139556099413398,0.909124934179681,0.996233212271177,0.855306841266058,0.909124934179681,0.879962617937312,0.993911405460269,0.00376678772882303,746,78.8888888888889,20924,54.7777777777778,800.777777777778,824.888888888889
"4","rnd","NJIT4321-rnd","brats_2013_pat0130_1","ablaze_gull",4.66666666666667,0.85306228612433,0.950463081424465,0.860901046931854,0.865114219533763,0.860857546901481,0.765289506462121,0.165902118593191,0.983989082060977,0.976890577030139,0.974069012629503,0.90775738368346,4.81983416902026,0.180602636858228,0.11416432081964,0.104953444577379,0.0149408163861948,0.15932446284558,0.905340924248747,0.995585238600183,0.849506729173741,0.905340924248747,0.865114219533763,0.991917129836001,0.00441476139981669,725.222222222222,89.2222222222222,20432.4444444444,81.1111111111111,806.333333333333,814.444444444444
"5","simple","simple","brats_2013_pat0130_1","ablaze_gull",5.33333333333333,0.85114266479814,0.948252440431362,0.858939084118719,0.863141736009654,0.858899132291815,0.762269269811805,0.165170605240332,0.984029969073529,0.978056597734505,0.975025528879729,0.910458708845115,4.24374989738083,0.179670438458937,0.107656330926193,0.104676641737853,0.0148917217218999,0.161945633247734,0.900627942177379,0.995876938685345,0.848419539019799,0.900627942177379,0.863141736009654,0.991938377123562,0.00412306131465471,719.888888888889,83.5555555555556,20438.1111111111,86.4444444444444,806.333333333333,803.444444444444
"6","zyx","zyx","brats_2013_pat0130_1","ablaze_gull",5,0.852937839717816,0.946405048878467,0.860405209999464,0.864421349815021,0.860343196696214,0.765679742754649,0.167425516242723,0.984621691861738,0.992079844226351,0.975692379970523,0.886529546039636,5.87456263780313,0.190822351731851,0.110378761504372,0.0990425203771905,0.0141114720044004,0.162026913526487,0.896740018774269,0.996070078982665,0.864244423545339,0.896740018774269,0.864421349815021,0.992240792064425,0.00392992101733526,722.333333333333,79.2222222222222,20442.4444444444,84,806.333333333333,801.555555555556
"7","rnd","UTintelligence-rnd","brats_2013_pat0134_1","ablaze_gull",1,0.439786545572739,0.734625155724745,0.465756360468916,0.488884490908871,0.46288397904837,0.347261286186172,0.104377530895259,0.907741141155794,0.765389141177261,0.654499477639931,0.696681027725238,21.7259079503732,2.41851089469065,0.46886831548909,0.401259375354614,0.0738450010093984,1.33323405520428,0.492371538620209,0.976878772829282,0.594188643079343,0.492371538620209,0.488884490908871,0.951254405933036,0.0231212271707184,736.444444444444,481.777777777778,19740.8888888889,574.555555555556,1311,1218.22222222222
"8","simple","simple","brats_2013_pat0134_1","ablaze_gull",3.33333333333333,0.785524625502044,0.901316639398279,0.801388538397059,0.810880384656072,0.801313556069467,0.696584302979966,0.207963331040685,0.965077426818364,0.958667243569809,0.928427262204787,0.916562197208537,12.6546214655814,0.472279577962782,0.107353421791921,0.197743216013271,0.0320484916833766,0.26517778145634,0.809476686023845,0.993156592772713,0.824707032687797,0.809476686023845,0.810880384656072,0.982142000911457,0.0068434072272868,1066.88888888889,141.222222222222,20081.4444444444,244.111111111111,1311,1208.11111111111
"9","zyx","zyx","brats_2013_pat0134_1","ablaze_gull",5.33333333333333,0.808816961290452,0.915389573957884,0.82255657191667,0.830680387649496,0.822524512693832,0.724253190335505,0.22263294830665,0.969975154484348,0.968502955091587,0.936460323238279,0.935112385680564,14.3628067945127,0.484728125133601,0.0985229007138266,0.177639450016539,0.0279718268170166,0.229407179518171,0.837724911061577,0.99305423685419,0.832290331137827,0.837724911061577,0.830680387649496,0.984701258781098,0.00694576314580967,1122.22222222222,144.555555555556,20078.1111111111,188.777777777778,1311,1266.77777777778
"10","rnd","Misfits-rnd","brats_2019_138_1","ablaze_gull",3,0.516106062592575,0.873715987224171,0.523038256992382,0.527656540346325,0.521661240623707,0.432121538305335,0.101222815104127,0.977103022412987,0.620103370493099,0.650702373280009,0.601390174958572,19.2537294363992,3.56344098531587,0.742348348696815,0.118109524669584,0.017378193265862,10.1585752147551,0.754093369661688,0.993338604786654,0.503577431529888,0.754093369661688,0.527656540346325,0.988377655513698,0.00666139521334549,447.111111111111,136.666666666667,20606,97.8888888888889,545,583.777777777778
"11","simple","simple","brats_2019_138_1","ablaze_gull",3.66666666666667,0.706452946686834,0.864953464263324,0.708649608736678,0.709924711751815,0.708580745993345,0.612548925090868,0.126104497349992,0.994785970651161,0.89537673459454,0.822033177974469,0.779958731884256,19.8501973757616,2.61846276424863,0.467761562013242,0.0385223535652684,0.00460614018952978,0.716819072659924,0.730891949753302,0.999014978773347,0.814645168472805,0.730891949753302,0.709924711751815,0.997383637222852,0.000985021226653005,510.777777777778,19.3333333333333,20723.3333333333,34.2222222222222,545,530.111111111111
"12","zyx","zyx","brats_2019_138_1","ablaze_gull",3.66666666666667,0.610753810401177,0.779712151049051,0.612727017856821,0.613869106450475,0.612524486488184,0.551363347584298,0,0.994766944943347,0,0.676414733802843,0.632006181002577,32.7153630434273,5.88313878476988,0.904099638796773,0,0.103402624869185,0.195759812968779,0.559908770106544,0.999515531991557,0.743154734248098,0.559908770106544,0.613869106450475,0.997373332439284,0.000484468008443052,500.444444444444,9.33333333333333,20733.3333333333,44.5555555555556,545,509.777777777778
"13","rnd","jaguars-rnd","brats_2019_141_1","ablaze_gull",4.33333333333333,0.776798511092125,0.9317995657587,0.788648121769615,0.795339119656737,0.788418290173286,0.679887517482747,0.196866342094433,0.974220482589075,0.872334990138839,0.900384822932791,0.847074704404622,16.5906961635536,0.919603987391872,0.123497536657694,0.155282955291875,0.023063324485323,0.298026651717319,0.870914581312203,0.992684550205198,0.777214433083584,0.870914581312203,0.795339119656737,0.986931119159787,0.00731544979480233,882.888888888889,136.555555555556,18413.5555555556,120.333333333333,1003.22222222222,1019.44444444444
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"16","rnd","FightGliomas-rnd","brats_2019_90_1","ablaze_gull",6,0.968257954413161,0.987093362528072,0.970800193935607,0.972048533746708,0.970801355739778,0.94570310089342,0.204682906269684,0.995299728580093,0.99851769262895,0.99773209120061,0.990706184810191,2.35337109861099,0.0344489741039471,0.0285055152008294,0.0397278812479393,0.00464629073617492,0.0288059740714348,0.975754720290816,0.998432004765327,0.9685629127278,0.975754720290816,0.972048533746708,0.997641026050083,0.00156799523467276,770.222222222222,30.7777777777778,20064.1111111111,18.5555555555556,788.777777777778,801
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"19","rnd","Tyagi-rnd","brats_MDA_945_1","ablaze_gull",4.33333333333333,0.745555569694423,0.885669742953136,0.759263828544658,0.767838981407406,0.759157345204569,0.633507916050858,0.146023656683058,0.967517603171903,0.881010728788507,0.88835140017712,0.910730480827375,12.3514245536757,0.815941012416867,0.280601413840585,0.188352804604093,0.0295388388785871,0.328332764553787,0.78016854749535,0.991170938410922,0.772205515354681,0.78016854749535,0.767838981407406,0.983443927479636,0.00882906158907816,717.111111111111,180.111111111111,20582.4444444444,175.666666666667,892.777777777778,897.222222222222
"20","simple","simple","brats_MDA_945_1","ablaze_gull",5.33333333333333,0.826356297027293,0.916198592408247,0.836785251821064,0.842660724647247,0.836755885662414,0.731791097441842,0.169311412198244,0.977814120503358,0.967413416696995,0.953399039161258,0.933051000360542,7.68751024370488,0.267137250532801,0.106993910988983,0.141276068501097,0.0208114845989645,0.191602981147392,0.838959573262704,0.993437611553791,0.857374950750217,0.838959573262704,0.842660724647247,0.98876460334073,0.00656238844620888,785.555555555556,131.888888888889,20630.6666666667,107.222222222222,892.777777777778,917.444444444444
edit
I somehow got a calculation running now by reducing some predictors and switching the family variable and getting rid of my random variables for the moment:
glm3 <- glmmLasso(stars ~ ADJRIND + AUC + KAPPA + DICE + ICCORR + JACRD + MUTINF + RNDIND + SURFOVLP + SURFDICE + VOLSMTY + HDRFDST + AVGDIST + MAHLNBS + VARINFO + GCOERR + PROBDST + SNSVTY + SPCFTY + PRCISON + RECALL + FMEASR + ACURCY + FALLOUT, rnd = NULL, family = cumulative(), data = mixedModel_df, lambda=10, switch.NR=TRUE, control=list(print.iter=TRUE))
However the calculation is stuck at iteration 89 for 30minutes now.
I have trouble deciding which one is the correct family type for my problem. This question is probably better suited for CV. How do I correctly specify my random variables, my final model will have two?

Simplify boolean equation from truth table

I need help simplifying the following to the simplest terms. Boolean algebra just doesn't quite click with me yet, any help is appreciated.
(!A!B!C)+(!AB!C)+(!ABC)+(A!B!C)+(A!BC)+(AB!C)
I got it to the following, but I don't know where to go from here:
!A(!B!C + B!C + BC) + A(!B!C + B(XOR)C)
If you are curious and want to check my previous work, I got the original equation from the truth table:

Initially we have A(~B~C + ~BC + ~CB) + ~A(~B~C + B~C + BC)
First Term: A(~B~C + ~BC + ~CB)
= A(~B(~C + C) + ~CB)
= A(~B(True) + ~CB)
= A(~B + ~CB)
= A((~B + ~C)(~B + B))
= A((~B + ~C)(True))
= A(~B + ~C)
Second Term: ~A(~B~C + B~C + BC)
= ~A(~C(~B + B) + BC)
= ~A(~C(True) + BC)
= ~A(~C + BC)
= ~A((~C + C) (~C + B))
= ~A((True) (~C + B))
= ~A(~C + B)
So First Term + Second Term becomes: ~A(~C + B) + A(~B + ~C)
= ~A~C + ~AB + A~B + A~C
= AxorB + ~A~C + A~C
= AxorB + ~C(~A + A)
= AxorB + ~C(True)
= AxorB + ~C
Hence we end up with AxorB + ~C

Simplify the following expression to A'D' + A'B

𝑓 = 𝐴'𝐵'𝐶'𝐷' + 𝐴'𝐵'𝐶𝐷' + 𝐴'𝐵𝐶'𝐷 + 𝐴'𝐵𝐶𝐷 + 𝐴'𝐵𝐶𝐷' + 𝐴'𝐵'𝐶𝐷' + 𝐴𝐵'𝐶D'
I've done the K-Map for the following and got A'D' + A'B but can't figure out how to show the simplification that the K-Map derived.

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