eDay

07022018, 06:24 PM
Post: #3




RE: eDay
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Hi, Patrick: First of all, welcome to the forum. I fully expect you'll enjoy the place and enrich us all with your own contributions. That said: Quote:Today I'm celebrating eDay. At least in the rest of the date format world a look at the calendar shows 2.718! Indeed, it shows in that order where I live. To contribute to your celebration, here's a very short program I wrote many many years ago to compute e to in excess of 200 digits in an HP15C. Here's an extract taken from my article "Long Live the HP15C !": This 64step program will compute from 8 to 208 decimal places of Euler’s constant, the wellknown transcendental number e = 2.71828+ . It is by no means optimized for performance but tries instead to be as short and straightforward as possible. Although you can compute more decimal places in an HP15C, for the purposes of this article this simpler program will do nicely. Program listing: Very Important: steps 30 and 43 must be entered in USER mode, while all the rest must be entered out of USER mode. An ulike character must appear next to the step number only for steps 30 and 43, and no others. 01 LBL A 23 RCL A 45 FRAC 02 MATRIX 0 24 X=0? 46 CHS 03 MATRIX 1 25 ISG 2 47 RTN 04 1 26 LBL 0 48 LBL 5 05 DIM A 27 RCL A 49 FRAC 06 DIM B 28 RCL÷ I 50 RCL B 07 RESULT B 29 INT 51 INT 08 STO 2 30u STO A 52 RCL RAN# 09 STO I 31 GTO 2 53 * 10 EEX 32 RCL I 54 + 11 8 33 PSE 55 R/S 12 STO A 34 RCL MATRIX A 56 GTO 4 13 1/X 35 MATRIX 7 57 LBL 2 14 STO RAN# 36 TEST 0 58 RCL* I 15 LBL 1 37 GTO 1 59  16 RCL MATRIX A 38 RCL MATRIX B 60 RCL RAN# 17 RCL MATRIX B 39 RCL RAN# 61 ÷ 18 + 40 * 62 STO+ A 19 RCL 2 41 FIX 8 63 RCL A 20 STO O 42 LBL 4 64 GTO 0 21 ISG I 43u RCL B 22 LBL 3 44 GTO 5 The constant e is computed using the wellknown formula: e = 1 + 1/1! +1/2! + 1/3! + 1/4! + ... using enough terms to achieve the desired precision. We use the powerful matrix capabilities of the HP15C, holding the current term in a vector (matrix A), and the running sum in another vector (matrix B). Steps 0114 dimension and initialize both matrices, as well as some indexes and ancillary constants. Starting with 1, each term is computed by dividing the previous one by the corresponding divisor up to a maximum of 208 decimal digits, until we arrive at a term that is zero to the specified precision. This multiprecision arithmetic is done by considering each term as composed of N blocks (126), each holding 8 digits, the extra digits in each register being carried out to the next block. Since we may use divisors up to 125, each block is limited to 8 digits, lest the carry would make the next block larger than the 10 digits an HP15C storage register can hold. The addition of each multiblock term to the running total is done using matrix arithmetic in steps 1618. Steps 1925 increment the divisor and keep track of the first nonzero block of each term to optimize speed by avoiding unnecesary operations. Steps 2631 and 5764 perform the multiprecision division. The process ends when the current term has all its blocks equal to zero. As all blocks always hold positive values, we can use an advanced matrix operation, the Row Norm, to test for finalization, because in this case the Row Norm will equal zero if and only if all block values are zero. This is tested in steps 3437. Once this condition is met, the result matrix which holds the running sum is scaled down for display using matrixscalar arithmetic (steps 3840). The computed answer is then displayed by recalling each block in turn, adding the carry from the previous block, and marking the last one negative (steps 4156). As you can see, though simple, this program does use some of the HP15C’s advanced programming capabilities (as well as a trick or two), including basic matrix operations (MATRIX 1), advanced matrix functions (MATRIX 7), recall arithmetic (RCL÷ I), using registers as indexes including incrementandtest operations (ISG 2) , autoincrementtestandloop matrix element access (uSTO A, uRCL B), matrix arithmetic (steps 18 and 40), etc. Usage instructions: 1) After keying in the program, make sure you’re not in complex mode (press CF 8 if in doubt), then you must commit enough storage registers to the common pool for the matrix operations. To that effect, press: 2, f DIM (i) 2) Now, enter the number of 8digit blocks you want to use, from 1 (8 digits) to 26 (208 digits). For example, if you want to compute 200 decimal digits of e, you must specify 200/8 = 25 blocks. Start the program by pressing either f PRGM, R/S or GSB A or (in User mode) A While running, it will briefly show each succesive divisor used (2, 3, ...), then once the computation is over, it will display each block of 8 decimal digits (with an initial “0.”, in order to preserve leading zeros at the left end of the block), starting with decimals 1st8th. The very last block will be marked negative, to signal the end of the output. Let’s see an example: to compute the first 24 decimal digits of e, we specify 24/8 = 3 blocks, and proceed like this (in USER mode): 3, A > (2.00000000) [divisor = 2] ... [after 2’26”] > (25.00000000) [divisor = 25] > 0.71828182 [decimals 1st 8th] R/S > 0.84590452 [decimals 9th16th] R/S > 0.35360274 [decimals 17th24th] so, after adding a “2.” at the front and writing down all 8digit blocks (that is, minus the initial “0.” or “0.”) in their proper order, we finally get: e = 2.71828182 84590452 35360274 where, due to the accumulation of rounding errors during the process, the last block of the computed answer comes out as “3536 0274” while correct is “3536 0287”, so we have an error of 13 ulps (units in the last place). 3) Should you need to display the output again, press: GSB 4 For 208 decimal digits, the final result is: e = 2.71828182 84590452 35360287 47135266 24977572 47093699 95957496 69676277 24076630 35354759 45713821 78525166 42742746 63919320 03059921 81741359 66290435 72900334 29526059 56307381 32328627 94349076 32338298 80753195 25101901 15738281 The last block of the computed answer comes out as “1573 8281” while correct is “1573 8341”, so the error is 60 ulps, and thus after rounding the last two places we’ve got 206 decimals fully correct. Regards. V. . All My Articles & other Materials here: Valentin Albillo's HP Collection 

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Messages In This Thread 
RE: eDay  Thomas Klemm  07022018, 01:57 PM
RE: eDay  Valentin Albillo  07022018 06:24 PM
RE: eDay  Valentin Albillo  07032018, 11:25 PM
RE: eDay  Thomas Klemm  07042018, 07:06 PM
RE: eDay  Thomas Klemm  07062018, 09:27 PM

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