In this post, I would like to explain how I have used Lambda to create a function to generate a Fibonacci series array. This example can also be used to understand how to create an array where th...
I have still to get to grips with your formula; I find recursive formulae especially opaque!
One observation, though. It is possible to conduct the recursion, and maintain a counter that could be used as a backup termination criterion, without building the array as you go.
When the end point is reached, all the values obtained along the way are still present within the memory stack and are accessible one at a time following return from the previous level. In the Fibonacci example I used this to combine the current value and the return array to build the complete series as an array of known length.
It's known that recursive lambda allows quite limited number of iteration. I tried to play with different variants of FIB() presented to find max allowed parameter for each. Actually that's about number of iteration, we are within max returned number limit.
In last case I just added text constant within LET().
That's not an analysis, just stats.
Resources
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This example can also be used to understand how to create an array where th...","body@stringLength":"6056","rawBody":"
In this post, I would like to explain how I have used Lambda to create a function to generate a Fibonacci series array.
This example can also be used to understand how to create an array where the value of an element in the array depends on previous elements in the same array.
Generating Fibonacci as a dynamic array by @Viz
This function has been created using three function in two layers. The inner layer functions include the following:
InFib: This function generates the Nth Fibonacci number
InFibSer: This function generates the entire Fibonacci series up to the Nth number
The two functions mentioned above require arguments that are complicated and less intuitive. Therefore, we use an outer layer function (FibSeries) that generates the necessary argument and calls the two functions to generate the series.
=InFIB
'This function returns nth Fibonacci using tail recursion 'It takes three arguments n, a=0 , b=1 as argument and returns the nth Fibonacci value
=LAMBDA(n,a,b,IF(n=1,a,IF(n=2,b,Infib(n-1,b,a+b))))\n/* The value for the arguments a and b should be passed as 0 and 1, respectively. */
Example:
=InFIB(10,0,1) would generate 34 as the value
Our objective, here, is not to obtain a single value but to generate an entire array. However, here is where we have a challenge: While it is easier to manipulate an existing array, it is far more to difficult to create a new array (other than sequence array) or extend an existing array.
Therefore, in order to create a Fibonacci series, we are going to pass a dummy array of n values. This array will have 0 as the first element and 1 as the value for all the other elements.
The following image portrays the recursive addition process that converts that the initial array into a Fibonacci series.
The following code does carries out the process recursively till the maximum value of the array equals the nth Fibonacci value.
=INFIBSER
'This inner function takes three arguments and returns a Fibonacci series 'The three arguments are nth Fibonacci value, the initial array and a sequence array (sequential numbers from 1 to n) 'Note, the sequence array can instead be generated inside the function but it may inefficient
In the first iteration, the above function adds one to all the values from the fourth position onwards. For the subsequent iterations it adds the second largest unique value to all values that all 2 places after the position of such second largest value. The above function uses the match function to identify the position pointer.
Although the above function can generate the Fibonacci series, as mentioned earlier, it requires arguments that are less intuitive and which requires some set up.
Therefore, here we have an outer layer function that requires only one argument, n. It creates the necessary argument to be passed to the inner layer function and calls them.
=FIBSERIES
'This outermost function generates an initial array, a sequence array, and nth fibonnaci value 'It then calls the inner functions with these parameters
This final function can be used by the user to generate the Fibonacci series by merely passing the number of values required as argument.
Hope you found this useful. I would love to know if you think there is a way to improve the algorithm, here. I am especially looking for suggestions that could replace the match / xmatch in the InFibSer function to get the position pointer. Since the positions move sequentially, I believe a mathematical solution may exist.
In this post, I would like to explain how I have used Lambda to create a function to generate a Fibonacci series array.
This example can also be used to understand how to create an array where the value of an element in the array depends on previous elements in the same array.
Generating Fibonacci as a dynamic array by @Viz
This function has been created using three function in two layers. The inner layer functions include the following:
InFib: This function generates the Nth Fibonacci number
InFibSer: This function generates the entire Fibonacci series up to the Nth number
The two functions mentioned above require arguments that are complicated and less intuitive. Therefore, we use an outer layer function (FibSeries) that generates the necessary argument and calls the two functions to generate the series.
=InFIB
'This function returns nth Fibonacci using tail recursion 'It takes three arguments n, a=0 , b=1 as argument and returns the nth Fibonacci value
=LAMBDA(n,a,b,IF(n=1,a,IF(n=2,b,Infib(n-1,b,a+b))))\n/* The value for the arguments a and b should be passed as 0 and 1, respectively. */
Example:
=InFIB(10,0,1) would generate 34 as the value
Our objective, here, is not to obtain a single value but to generate an entire array. However, here is where we have a challenge: While it is easier to manipulate an existing array, it is far more to difficult to create a new array (other than sequence array) or extend an existing array.
Therefore, in order to create a Fibonacci series, we are going to pass a dummy array of n values. This array will have 0 as the first element and 1 as the value for all the other elements.
The following image portrays the recursive addition process that converts that the initial array into a Fibonacci series.
The following code does carries out the process recursively till the maximum value of the array equals the nth Fibonacci value.
=INFIBSER
'This inner function takes three arguments and returns a Fibonacci series 'The three arguments are nth Fibonacci value, the initial array and a sequence array (sequential numbers from 1 to n) 'Note, the sequence array can instead be generated inside the function but it may inefficient
In the first iteration, the above function adds one to all the values from the fourth position onwards. For the subsequent iterations it adds the second largest unique value to all values that all 2 places after the position of such second largest value. The above function uses the match function to identify the position pointer.
Although the above function can generate the Fibonacci series, as mentioned earlier, it requires arguments that are less intuitive and which requires some set up.
Therefore, here we have an outer layer function that requires only one argument, n. It creates the necessary argument to be passed to the inner layer function and calls them.
=FIBSERIES
'This outermost function generates an initial array, a sequence array, and nth fibonnaci value 'It then calls the inner functions with these parameters
This final function can be used by the user to generate the Fibonacci series by merely passing the number of values required as argument.
Hope you found this useful. I would love to know if you think there is a way to improve the algorithm, here. I am especially looking for suggestions that could replace the match / xmatch in the InFibSer function to get the position pointer. Since the positions move sequentially, I believe a mathematical solution may exist.
It's known that recursive lambda allows quite limited number of iteration. I tried to play with different variants of FIB() presented to find max allowed parameter for each. Actually that's about number of iteration, we are within max returned number limit.
\n
\n
In last case I just added text constant within LET().
\n
That's not an analysis, just stats.
","body@stringLength":"1010","rawBody":"
\n
It's known that recursive lambda allows quite limited number of iteration. I tried to play with different variants of FIB() presented to find max allowed parameter for each. Actually that's about number of iteration, we are within max returned number limit.
\n
\n
In last case I just added text constant within LET().
\n
That's not an analysis, just stats.
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SergeiBaklan Good point. I think FIB should probably truncate decimals to integers if entered as parameters to be consistent with other functions. It seems not much change is required to do that because SEQUENCE and INDEX functions already do this (though to be precise one should check values very close to integer as you say.)
Viz I found this pretty challenging and don't find recursion very intuitive either. I first laid out the expected results for all input values from 1 to 10 horizontally referring to prior results using spill operator and then substituted LET to make it more efficient.
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Good point. I think FIB should probably truncate decimals to integers if entered as parameters to be consistent with other functions. It seems not much change is required to do that because SEQUENCE and INDEX functions already do this (though to be precise one should check values very close to integer as you say.)
I found this pretty challenging and don't find recursion very intuitive either. I first laid out the expected results for all input values from 1 to 10 horizontally referring to prior results using spill operator and then substituted LET to make it more efficient.
lori_mWas struggling to figure out how to extend an array as well, thanks for this example. Using your FIB array extension as a hint I've got a generic iterator layout after struggling with the formula a bit. Formula was returning a #NUM error, and figured out that TYPE(INDEX(Seed,ROWS(Seed)))=64 (array) rather than 1 for number. Any ideas on better way to do this? Decided to work through Project Euler problems to wrap my head around LAMBDA usage and this is part of my solution to question on Collatz sequence.
Was struggling to figure out how to extend an array as well, thanks for this example. Using your FIB array extension as a hint I've got a generic iterator layout after struggling with the formula a bit. Formula was returning a #NUM error, and figured out that TYPE(INDEX(Seed,ROWS(Seed)))=64 (array) rather than 1 for number. Any ideas on better way to do this? Decided to work through Project Euler problems to wrap my head around LAMBDA usage and this is part of my solution to question on Collatz sequence.
I have still to get to grips with your formula; I find recursive formulae especially opaque!
One observation, though. It is possible to conduct the recursion, and maintain a counter that could be used as a backup termination criterion, without building the array as you go.
When the end point is reached, all the values obtained along the way are still present within the memory stack and are accessible one at a time following return from the previous level. In the Fibonacci example I used this to combine the current value and the return array to build the complete series as an array of known length.
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I have still to get to grips with your formula; I find recursive formulae especially opaque!
One observation, though. It is possible to conduct the recursion, and maintain a counter that could be used as a backup termination criterion, without building the array as you go.
When the end point is reached, all the values obtained along the way are still present within the memory stack and are accessible one at a time following return from the previous level. In the Fibonacci example I used this to combine the current value and the return array to build the complete series as an array of known length.
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