Consider the following two row Vectors:

$X≔\mathrm{Vector}\left['\mathrm{row}'\right]\left(\left[1\,2\,3\,4\,5\,4\,3\,2\,1\right]\right)$

$X≔\left[\begin{array}{ccccccccc}1& 2& 3& 4& 5& 4& 3& 2& 1\end{array}\right]$

$Y≔\mathrm{Vector}\left['\mathrm{row}'\right]\left(\left[10\,20\,30\,40\,50\,60\,70\,80\,90\right]\right)$

$Y≔\left[\begin{array}{ccccccccc}10& 20& 30& 40& 50& 60& 70& 80& 90\end{array}\right]$

What values in $Y$ occur at the same places that 3 appears in $X$?

$\mathrm{Lookup}\left(3\,X\,Y\right)$

$30,70$

If we instead search from right to left:

$\mathrm{Lookup}\left(3\,X\,Y\,'\mathrm{direction}'='\mathrm{reverse}'\right)$

$70,30$

The number of results can be specified, in both the forward and reverse directions:

$\mathrm{Lookup}\left(3\,X\,Y\,'\mathrm{numresults}'=1\right)$

$30$

$\mathrm{Lookup}\left(3\,X\,Y\,'\mathrm{direction}'='\mathrm{reverse}'\,'\mathrm{numresults}'=1\right)$

$70$

Consider the following Matrix:

$A≔\mathrm{Matrix}\left(\left[\left[a\,1\,4\,7\right]\,\left[b\,2\,5\,8\right]\,\left[c\,3\,6\,9\right]\right]\right)$

$A≔\left[\begin{array}{cccc}a& 1& 4& 7\\ b& 2& 5& 8\\ c& 3& 6& 9\end{array}\right]$

If we don't specify the orientation (row or column), match container, and lookup container, it is assumed the match container is the second column and the lookup container is the first column:

$\mathrm{Lookup}\left(2\,A\right)$

$b$

Indexing from both the left and right are recognized:

$\mathrm{Lookup}\left(7\,A\,'\mathrm{column}'\,4\,1\right)$

$a$

$\mathrm{Lookup}\left(7\,A\,'\mathrm{column}'\,-1\,1\right)$

$a$

We can also specify rows as the match and lookup containers:

$\mathrm{Lookup}\left(6\,A\,'\mathrm{row}'\,3\,1\right)$

$4$

Exact matches for real and complex data may require floats:

$U≔\left[\mathrm{sqrt}\left(2\right)\,\mathrm{evalf}\left(\mathrm{sqrt}\left(2\right)\right)\,\mathrm{sqrt}\left(3\right)\,\mathrm{evalf}\left(\mathrm{sqrt}\left(3\right)\right)+\mathrm{Float}\left(2\,1-\mathrm{Digits}\right)\right]$

$U≔\left[\sqrt{2}\,1.414213562\,\sqrt{3}\,1.732050810\right]$

$V≔\left[''a''\,''b''\,''c''\,''d''\right]$

$V≔\left[''a''\,''b''\,''c''\,''d''\right]$

We cannot detect the numerical approximation for $\sqrt{2}$ without floats, but with floats and the default options we can detect it:

$\mathrm{Lookup}\left(\mathrm{sqrt}\left(2\right)\,U\,V\right)$

$''a''$

$\mathrm{Lookup}\left(\mathrm{sqrt}\left(2\right)\,U\,V\,'\mathrm{match}'='\mathrm{float}'\right)$

$''a'',''b''$

For the perturbed numerical approximation of $\sqrt{3}$, on the other hand, we will need to use floats with more ULPs:

$\mathrm{Lookup}\left(\mathrm{sqrt}\left(3\right)\,U\,V\right)$

$''c''$

$\mathrm{Lookup}\left(\mathrm{sqrt}\left(3\right)\,U\,V\,'\mathrm{match}'='\mathrm{float}'\right)$

$''c''$

$\mathrm{Lookup}\left(\mathrm{sqrt}\left(3\right)\,U\,V\,'\mathrm{match}'='\mathrm{float}'\,'\mathrm{ulp}'=2\right)$

$''c'',''d''$

Consider the following two lists:

$A≔\left[''ab''\,''abc''\,''ad''\,''abbc''\,''a2c''\,''aBc''\right]$

$A≔\left[''ab''\,''abc''\,''ad''\,''abbc''\,''a2c''\,''aBc''\right]$

$B≔\left[10\,20\,30\,40\,50\,60\right]$

$B≔\left[10\,20\,30\,40\,50\,60\right]$

Matches with strings can be detected using regular expressions. For example, let's take the match value to be the regular expression for a string starting with "a", ending with "c", and having one or more characters, all lowercase letters, between them:

$\mathrm{Lookup}\left(''(^)(a)([a-z]+)(c)(\$)''\,A\,B\,'\mathrm{match}'='\mathrm{regexp}'\right)$

$20,40$

We can also use wildcards:

$\mathrm{Lookup}\left(''a*c''\,A\,B\,'\mathrm{match}'='\mathrm{wildcard}'\right)$

$20,40,50,60$

The Lookup command can also accept custom matching procedures to, for example, search for values in matchdata that are less than matchvalue:

$\mathrm{Lookup}\left(4\,⟨1\,2\,3\,4\,5⟩\,⟨6\,7\,8\,9\,10⟩\,'\mathrm{match}'='\mathrm{less\_than}'\right)$

$6,7,8$

When match is a list with the first element being a procedure, the additional terms in the list are passed as additional arguments to the procedure:

$F≔\mathrm{Vector}\left['\mathrm{row}'\right]\left(\left[\mathrm{seq}\left(1..26\right)\right]\right)$

$F≔\left[\begin{array}{cccccccccccccccccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15& 16& 17& 18& 19& 20& 21& 22& 23& 24& 25& 26\end{array}\right]$

$G≔\mathrm{Vector}\left['\mathrm{row}'\right]\left(\left[\mathrm{seq}\left(''a''..''z''\right)\right]\right)$

$G≔\left[\begin{array}{cccccccccccccccccccccccccc}''a''& ''b''& ''c''& ''d''& ''e''& ''f''& ''g''& ''h''& ''i''& ''j''& ''k''& ''l''& ''m''& ''n''& ''o''& ''p''& ''q''& ''r''& ''s''& ''t''& ''u''& ''v''& ''w''& ''x''& ''y''& ''z''\end{array}\right]$

$\mathrm{Lookup}\left(20\,F\,G\,'\mathrm{match}'=\left[\mathrm{verify}\,'\mathrm{neighbourhood}'\left(5\,'\mathrm{open}'\right)\right]\right)$

$''p'',''q'',''r'',''s'',''t'',''u'',''v'',''w'',''x''$

$\mathrm{Lookup}\left(20\,F\,G\,'\mathrm{match}'=\left[\mathrm{verify}\,'\mathrm{neighbourhood}'\left(5\,'\mathrm{closed}'\right)\right]\right)$

$''o'',''p'',''q'',''r'',''s'',''t'',''u'',''v'',''w'',''x'',''y''$

Consider the following 2-D Array:

$A≔\mathrm{Array}\left(0..3\,0..2\,\left[\left[24\,11\,14\right]\,\left[10\,15\,13\right]\,\left[1\,13\,18\right]\,\left[5\,25\,9\right]\right]\right)$

Due to the column indices being indexed $0..2$ as opposed to $1..3$, the column index 2 can refer to both column 2 (absolute index) and column 3 (relative index). To indicate precisely what type of indices you are using with the Lookup command when there is ambiguity, you can pass the indices option:

$\mathrm{Lookup}\left(13\,A\,'\mathrm{column}'\,2\,1\,'\mathrm{indices}'='\mathrm{absolute}'\right)$

$15$

$\mathrm{Lookup}\left(13\,A\,'\mathrm{column}'\,2\,1\,'\mathrm{indices}'='\mathrm{relative}'\right)$

$1$

In this example, we work with a DataFrame of batting statistics for the 2021 Toronto Blue Jay hitters with 100 or more at bats:

$\mathrm{Data}≔\mathrm{Matrix}\left(\left[\left[0.242\,165\,19\,40\,8\,0\,8\,24\,19\,0.328\,0.436\right]\,\left[0.298\,640\,121\,191\,30\,1\,29\,102\,40\,0.343\,0.484\right]\,\left[0.224\,250\,27\,56\,10\,1\,7\,27\,37\,0.322\,0.356\right]\,\left[0.282\,131\,16\,37\,6\,2\,4\,15\,9\,0.329\,0.45\right]\,\left[0.223\,184\,32\,41\,13\,0\,11\,28\,17\,0.299\,0.473\right]\,\left[0.264\,299\,59\,79\,19\,1\,22\,50\,37\,0.352\,0.555\right]\,\left[0.246\,114\,9\,28\,6\,0\,2\,11\,8\,0.293\,0.351\right]\,\left[0.276\,500\,62\,138\,28\,2\,21\,84\,32\,0.319\,0.466\right]\,\left[0.265\,652\,115\,173\,39\,2\,45\,102\,66\,0.334\,0.538\right]\,\left[0.241\,511\,59\,123\,25\,1\,22\,81\,27\,0.281\,0.423\right]\,\left[20.53\,198\,22\,50\,15\,0\,1\,10\,15\,0.31\,0.343\right]\,\left[0.209\,139\,12\,29\,4\,1\,4\,8\,9\,0.272\,0.338\right]\,\left[0.311\,222\,32\,69\,13\,1\,2\,17\,22\,0.376\,0.405\right]\,\left[0.296\,550\,92\,163\,29\,0\,32\,116\,36\,0.346\,0.524\right]\,\left[0.311\,604\,123\,188\,29\,1\,48\,111\,86\,0.401\,0.601\right]\right]\right)\:$

$\mathrm{Players}≔\left[''Alejandro\; Kirk''\,''Bo\; Bichette''\,''Cavan\; Biggio''\,''Corey\; Dickerson''\,''Danny\; Jansen''\,''George\; Springer''\,''Joe\; Panik''\,''Lourdes\; Gurriel\; Jr''\,''Marcus\; Semien''\,''Randal\; Grichuk''\,''Reese\; McGuire''\,''Rowdy\; Tellez''\,''Santiago\; Espinal''\,''Teoscar\; Hernandez''\,''Vladimir\; Guerrero\; Jr''\right]\:$

$\mathrm{Categories}≔\left[''AVG''\,''AB''\,''R''\,''H''\,''2B''\,''3B''\,''HR''\,''RBI''\,''BB''\,''OBP''\,''SLG''\right]\:$

$\mathrm{DF}≔\mathrm{DataFrame}\left(\mathrm{Data}\,'\mathrm{rows}'=\mathrm{Players}\,'\mathrm{columns}'=\mathrm{Categories}\right)$

Recall that, for a DataFrame, if no lookup column is specified, the row labels are used.

Which players hit 30 or more home runs?

$\mathrm{Lookup}\left(30\,\mathrm{DF}\,''HR''\,'\mathrm{match}'='\mathrm{greater\_equal}'\right)$

$''Marcus\; Semien'',''Teoscar\; Hernandez'',''Vladimir\; Guerrero\; Jr''$

Which players hit 0.300 or higher and had 500 or more at bats?

$\mathrm{select}\left(\mathrm{member}\,\mathrm{Lookup}\left(0.300\,\mathrm{DF}\,''AVG''\,'\mathrm{match}'='\mathrm{greater\_equal}'\,'\mathrm{output}'='\mathrm{list}'\right)\,\mathrm{Lookup}\left(500\,\mathrm{DF}\,''AB''\,'\mathrm{match}'='\mathrm{greater\_equal}'\,'\mathrm{output}'='\mathrm{list}'\right)\right)$

$\left[''Vladimir\; Guerrero\; Jr''\right]$

Teoscar Hernandez led the Blue Jays with 116 RBIs in 2021. Who were the (at most) top five players that drove in 100 or more runs?

$\mathrm{Lookup}\left(100\,\mathrm{sort}\left(\mathrm{DF}\,''RBI''\right)\,''RBI''\,'\mathrm{match}'='\mathrm{greater\_equal}'\,'\mathrm{direction}'='\mathrm{reverse}'\,'\mathrm{numresults}'=1..5\right)$

$''Teoscar\; Hernandez'',''Vladimir\; Guerrero\; Jr'',''Marcus\; Semien'',''Bo\; Bichette''$