How to solve a set of Linear equations?

Question

I need to solve a system of linear equations:

aQ + bP = c
dQ + eP = f

Where:

a ~ N(100;10)
b ~ N(-1;0.1)
c ~ N(10;1)
d ~ N(10;0         


        

Answer 1

Just to offer another possible approach, you could use mapply() to effectively iterate over each coefficient and RHS vector in parallel:

set.seed(1);
a <- rnorm(100,100,10);
b <- rnorm(100,-1,0.1);
c <- rnorm(100,10,1);
d <- rnorm(100,10,0.1);
e <- rnorm(100,100,10);
f <- rnorm(100,10,0.1);
mapply(function(a,b,c,d,e,f) solve(matrix(c(a,d,b,e),2),matrix(c(c,f),2)),a,b,c,d,e,f);
##            [,1]       [,2]       [,3]      [,4]       [,5]       [,6]       [,7]       [,8]       [,9]      [,10]      [,11]      [,12]      [,13]      [,14]      [,15]      [,16]      [,17]      [,18]      [,19]      [,20]      [,21]      [,22]      [,23]     [,24]      [,25]      [,26]      [,27]      [,28]      [,29]      [,30]     [,31]      [,32]      [,33]      [,34]      [,35]      [,36]      [,37]      [,38]      [,39]      [,40]      [,41]      [,42]      [,43]     [,44]      [,45]      [,46]      [,47]      [,48]      [,49]     [,50]      [,51]      [,52]      [,53]     [,54]      [,55]      [,56]      [,57]     [,58]      [,59]      [,60]      [,61]      [,62]      [,63]      [,64]      [,65]     [,66]      [,67]      [,68]      [,69]      [,70]      [,71]     [,72]      [,73]      [,74]      [,75]     [,76]      [,77]      [,78]      [,79]      [,80]      [,81]      [,82]      [,83]      [,84]      [,85]      [,86]      [,87]      [,88]     [,89]     [,90]     [,91]      [,92]      [,93]      [,94]      [,95]      [,96]      [,97]      [,98]      [,99]     [,100]
## [1,] 0.11195915 0.11550647 0.12751804 0.0841975 0.07569079 0.13696420 0.10252485 0.09894877 0.09515799 0.10922284 0.08629273 0.10113763 0.10308528 0.11216529 0.09963416 0.11659483 0.09791256 0.08081292 0.09926686 0.09472402 0.07648024 0.09355051 0.09391840 0.1219382 0.08409615 0.11939194 0.09931966 0.09958410 0.10817085 0.09947289 0.0803567 0.07280706 0.09108175 0.10730635 0.11623282 0.10439468 0.11082056 0.08964156 0.10089985 0.09365293 0.10998488 0.11389442 0.09656601 0.0872372 0.12093390 0.08759978 0.09202209 0.09137391 0.10043375 0.1025373 0.09823991 0.11186173 0.09698966 0.1112308 0.09463053 0.09386283 0.07966344 0.1193879 0.09922995 0.09776010 0.08903184 0.09724774 0.09156914 0.10903049 0.12760021 0.1014739 0.11805636 0.07770935 0.09601201 0.07510938 0.09370570 0.1130247 0.08705415 0.14043486 0.11704624 0.1091617 0.08154497 0.10810454 0.08690775 0.11685334 0.11133037 0.09817079 0.10215157 0.11097263 0.08957527 0.08781713 0.08506901 0.11398943 0.1017468 0.1080845 0.1026900 0.09321191 0.09276108 0.08089681 0.10226763 0.09705664 0.12423443 0.11736748 0.11432446 0.10282015
## [2,] 0.08016778 0.07420647 0.09132015 0.0953508 0.09734921 0.09111316 0.09172938 0.09253545 0.08018053 0.09452145 0.09233637 0.08349215 0.07979583 0.09634328 0.09443451 0.08404651 0.08153545 0.09468065 0.10482579 0.07829014 0.08002979 0.09773864 0.09023393 0.1070831 0.08657002 0.07433625 0.10757136 0.09891601 0.09612082 0.09671603 0.1139721 0.08951823 0.10746614 0.08889339 0.08501627 0.09050828 0.08181816 0.09255519 0.09681833 0.08489033 0.09251537 0.07567294 0.09099927 0.0841442 0.08769969 0.13057528 0.10425570 0.09546615 0.08564604 0.1174630 0.08258592 0.09420732 0.09640516 0.1048073 0.10757819 0.08931113 0.08544322 0.1124914 0.09978848 0.08516907 0.09438840 0.07408315 0.08145526 0.08357922 0.08661255 0.0884171 0.09454044 0.08658720 0.07824592 0.08435285 0.09760972 0.1068059 0.09319386 0.08188822 0.07865432 0.1037524 0.08916170 0.08914854 0.07982980 0.08718701 0.09064425 0.10193109 0.09645453 0.09396509 0.07402690 0.07399806 0.09308106 0.09940533 0.1126132 0.0841854 0.1002329 0.07365820 0.10095744 0.08985023 0.06348183 0.10252098 0.08397957 0.09875275 0.08654524 0.09736881

Or if you prefer getting the results as a list, you could pass SIMPLIFY=F to mapply(), or just use Map(), which doesn't simplify by default:

Map(function(a,b,c,d,e,f) solve(matrix(c(a,d,b,e),2),matrix(c(c,f),2)),a,b,c,d,e,f);
## [[1]]
##            [,1]
## [1,] 0.11195915
## [2,] 0.08016778
##
## [[2]]
##            [,1]
## [1,] 0.11550647
## [2,] 0.07420647
##
## ... (snip) ...
##
## [[99]]
##            [,1]
## [1,] 0.11432446
## [2,] 0.08654524
##
## [[100]]
##            [,1]
## [1,] 0.10282015
## [2,] 0.09736881

Answer 2

You could try using apply()

# data  
df = data.frame(a,b,c,d,e,f)

# apply function
out = t(apply(df, 1, 
      function(x){ 
      coefs = matrix( c(x['a'], x['d'], x['b'], x['e']), 2, 2); 
      ys = array(c(x['c'],x['f']),2); 
      out = solve(coefs, ys); 
      names(out) = c('P','Q');
      out}))

Or else using sapply()

out = t(sapply(seq(100), function(i){
       coefs = matrix(c(a[i], d[i], b[i] ,e[i]),2,2); ys = array(c(c[i],f[i]),2);
       out = solve(coefs, ys); names(out) = c('P','Q'); 
       out}))

and if you want to use for loop, here is what you could do

# declare matrix to store the output
out = matrix(ncol=2, nrow=100) 

# populate declared matrix using for loop
for(i in 1:100){  
    coefs = matrix(c(a[i],d[i],b[i],e[i]),2,2)
    ys = array(c(c[i],f[i]),2)
    out[i,] = as.vector(solve(coefs, ys))
    colnames(out) = c('P','Q')
    out}