ALARGESUM

[Xlambda]

ALARGESUM !! recursive !! sums vertical array of large numbers, calls LARGESUM

=LAMBDA(a,
    LET(n,ROWS(a),x,INDEX(a,n),
       IF(n=1,a,LARGESUM(ALARGESUM(INDEX(a,SEQUENCE(n-1))),x))
    )
)

LAMBDA 6.0.xlsx
	A	B
1	67834563234560684256775322456780876839
2	890746476745421097897675645434623145856
3	765543422314343656765675858532125340988
4	=ALARGESUM(A1:A3)
5	1724124462294325438920126826423529363683
6
ALARGESUM post

Cell Formulas
Range	Formula
A4	A4	=FORMULATEXT(A5)
A5	A5	=ALARGESUM(A1:A3)

 

[Xlambda]

LARGE NR OP.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L
1	Chapter 1. ADDition. The functions. Part 4. Last part of ALSUM
2	…,CONCAT(FILTER(d,SEQUENCE(,COLUMNS(d))>=XMATCH(TRUE,d<>0)))))
3
4		(to simulate more leading columns with 0's)
5		let's imagine an ipothetical "u" delivered by first part of ALSUM
6		48	143	1345	2453	976	4321
7
8		"d" variable (T_LA) will return this result:
9		=T_LA(B6:G6)
10		0	0	0	7	8	3	4	3	8	1
11		as we expected, row array has to have single digits values
12		(exit condition of recursive T_LS is met)
13
14		index position of first value <>0
15		=XMATCH(TRUE,B10#<>0)
16		4
17
18		=FILTER(B10#,SEQUENCE(,COLUMNS(B10#))>=B16)
19		7	8	3	4	3	8	1
20
21		=CONCAT(B19#)
22		7834381
23
ALADD 6

Cell Formulas
Range	Formula
B9,B21,B18,B15	B9	=FORMULATEXT(B10)
B10:K10	B10	=T_LA(B6:G6)
B16	B16	=XMATCH(TRUE,B10#<>0)
B19:H19	B19	=FILTER(B10#,SEQUENCE(,COLUMNS(B10#))>=B16)
B22	B22	=CONCAT(B19#)
Dynamic array formulas.

 

[Xlambda]

Chapter 1. ADDition.. Part 4. Examples, numbers as numbers (max 15 digits)
Introducing the iteration tool functions, I_T_LA and I_ALA, "designed" only to calculate total nr. of iterations performed by a specific calculation.
For that, we have to introduce a counter argument "i".
I_T_LA(a,[i ]) !!recursive!!
The only modifications made to main T_LA:
- from =LAMBDA(a,LET… to =LAMBDA(a,[i ],LET...
- from … IF(AND(LEN(u)=1),u,T_LA(u)))) to … IF(AND(LEN(u)=1),i,I_T_LA(u,i+1))))

=LAMBDA(a,[i ],
    LET(c,COLUMNS(a),l,LEN(a),m,MAX(l),r,SEQUENCE(m),s,SEQUENCE(,c+m-1)-r+1,
       u,MMULT(SEQUENCE(,m)^0,--IFERROR(INDEX(MID(REPLACE(a,1,0,REPT(0,m-l)),r+SEQUENCE(,c)^0-1,1),r,IF(s<1,-1,s)),0)),
      IF(AND(LEN(u)=1),i,I_T_LA(u,i+1))
    )
)

I_ALA(a)
Modifications made to main ALADD:
- removing last part, we need only to integrate I_T_LA that needs the values delivered by first part of ALADD
- from … d,T_LA(u) … to … I_T_LA(u)))

=LAMBDA(a,
    LET(l,LEN(a),m,MAX(l),u,MMULT(SEQUENCE(,ROWS(a))^0,--MID(REPLACE(a,1,0,REPT(0,m-l)),SEQUENCE(,m),1)),
      I_T_LA(u)
    )
)

Believe it or not, adding 1M numbers with max 15 digits was solved with only 4 (four) iterations !!!

LARGE NR OP.xlsx
	A	B	C	D	E	F	G	H	I	J	K	L	M	N
1	Chapter 1. ADDition.. Part 4. Examples, numbers as numbers (max 15 digits)
2
3	generating random array of numbers, 1M rows
4	between 1 and 999999999999999 (15 digits)
5	(beneath exponential notations, numbers are not modified)										nr. iterations
6	=RANDARRAY(1000000,,1,999999999999999,1)						=ALADD(A7#)				=I_ALA(A7#)
7	9.96681E+13						500067776293909745561				4
8	6.68401E+14
9	6.10809E+14		len check		average len		result len		Note:
10	9.69488E+14		=LEN(A7#)		check		=LEN(G7)		How iterations are counted.
11	4.78344E+13		14		=AVERAGE(C11#)		21		If exit condition is met from the first
12	5.75494E+14		15		14.88935				time, will have 0 iterations.
13	8.92371E+14		15
14	4.36856E+14		15					Examples		=ALADD(H15:H16)		=I_ALA(H15:H16)
15	4.08199E+14		14					1234		2468		0
16	7.97559E+14		15					1234
17	1.94442E+14		15
18	8.95554E+14		15					example used in post #9, we saw why another iteration was needed.
19	2.05022E+14		15
20	6.1971E+14		15							=ALADD(H21:H33)		=I_ALA(H21:H33)
21	2.11739E+14		15					99999		1110987		1
22	2.77154E+13		15					99999
23	3.3819E+13		15					9999
24	5.12099E+14		15					99999
25	7.47727E+14		15					99999
26	4.39849E+14		14					99999
27	4.6711E+14		14					99999
28	8.85025E+14		15					99999
29	1.47571E+13		15					999
30	7.86417E+14		15					99999
31	2.01398E+14		15					99999
32	4.4113E+14		15					99999
33	2.14276E+14		14					99999
34	5.46002E+14		15
ALADD 7

Cell Formulas
Range	Formula
A6,J20,L20,J14,L14,E11,C10,G10,K6,G6	A6	=FORMULATEXT(A7)
A7:A1000006	A7	=RANDARRAY(1000000,,1,999999999999999,1)
G7	G7	=ALADD(A7#)
K7	K7	=I_ALA(A7#)
C11:C1000010	C11	=LEN(A7#)
G11	G11	=LEN(G7)
E12	E12	=AVERAGE(C11#)
J15	J15	=ALADD(H15:H16)
L15	L15	=I_ALA(H15:H16)
J21	J21	=ALADD(H21:H33)
L21	L21	=I_ALA(H21:H33)
Dynamic array formulas.

 

[Xlambda]

Chapter 1. ADDition. Part 5. Examples, numbers as strings.
1M rows, 50 digits numbers.
Nr. iterations 3

LARGE NR OP.xlsx
	A	B	C	D	E	F	G	H
1	Chapter 1. ADDition. Part 5. Examples, numbers as strings
2
3	random array of numbers, 50 digits, 1M rows
4	=BYROW(RANDARRAY(1000000,50,0,9,1),LAMBDA(x,CONCAT(x)))		=ALADD(A5#)
5	19821742986165680477458318127292701722987146078094		49988206234824628070602175557121321759633237928595662293
6	95348550315098019725363199589459633534969227347501
7	00627104782103391832984073464956749875534309676028		nr. iterations		result len
8	28911941188555640635450040775311475425235457158017		=I_ALA(A5#)		=LEN(C5)
9	86906816290060039038636001401935929483366617765519		3		56
10	94666968314339971675656000542728545957901426340368
11	13380871283991771749147968054581593347843564880545		rows check
12	64477322276764662153686090675164610831275186383181		=ROWS(A5#)
13	64923878129329344680924392189326011498245203767817		1000000
14	69413199382731927281292711250134892171524433232757
15	62689997091355656367192864006272437701841509748826		numbers len check
16	88592749080021140142714432552863661120331570554027		=AND(LEN(A5#)=50)
17	43467642659113190252628006804183133967381187847541		TRUE
18	14560955019744129014940219541253064606160306925282
19	45117477326386556947357971711532243415901064932036
20	71254543056079355129497389136299207872480778332683
21	33843742905297297728068163098183140984703119865601
22	15663878016600355236226913285377862918431906302694
23	48307670456976811146570850421359155679370529024146
24	75510710250773875519128792820981083113366045185222
25	66972319912190801914404484188762843806904138030052
26	37654993480371823595664405498548178222461998669710
27	63881648612567777678055240663685725659913080001911
28	90459926509463409753073863960270321309323409166864
29	80538566540881965762261191965055619464876471684823
30	93665383666285704177596244982566197778907804967254
31	87356468053599081960116099578853065305820309738859
32	55327190655624216538648016546027342913057907262830
33	45814101372067974657425504237026891278139328538285
34	91789270070393441841335194345298908834703199531663
35	43383416781900773514115184842659724980451437015339
36	73928521791098336646413154313659083655156235123068
37	48713395928297635588428770379619267765230523285436
38	35991671627959637667121813721296526961332505492072
39	41353886215085184869095145308314999580548707792007
40	92364649557457749675807775860380961175753606652878
41	22225129293651468795374977730744084437834004782152
42	29949261245525433839604279709640538224943851879404
43	54583402208403779154219261103535503943279981589798
44	20199253826600868919040903951049904875350268984909
45	61676927657781961566645589946156540201130925201155
46	96186714291315817823007035161425744165590185729963
47	58843706181380281107633728249629220585238170944241
48	15929295892790348143365912155741920220425275776840
49	65299801393525523135542109312829750780066291740236
50	09199918120609031202708715663292212687069300222447
ALADD 8

Cell Formulas
Range	Formula
C4,C16,C12,E8,C8,A4	C4	=FORMULATEXT(C5)
A5:A1000004	A5	=BYROW(RANDARRAY(1000000,50,0,9,1),LAMBDA(x,CONCAT(x)))
C5	C5	=ALADD(A5#)
C9	C9	=I_ALA(A5#)
E9	E9	=LEN(C5)
C13	C13	=ROWS(A5#)
C17	C17	=AND(LEN(A5#)=50)
Dynamic array formulas.