Mathematics · Calculus
Newton Square-Root Iteration Calculator
Perform one Newton iteration toward the square root of a positive number.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- x_next=(3+10÷3)÷2=3.166666666666667.
- Squared error changes from -1 to 0.027777777777780344.
Understand Newton square-root step
One idea, three depths
Choose how deeply to explain Newton square-root step
Newton square-root step: Perform one Newton iteration toward the square root of a positive number.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Newton square-root step to answer this question: perform one newton iteration toward the square root of a positive number? Enter Target S and Current guess x; the calculator shows Next approximation. For example: For S=10 and guess 3, the next guess is 3.1667. The answer tells you Next approximation.
Age 15Explain it to a 15-year-oldConnect it to the formula
Newton's method replaces the current guess by the tangent line's root. The rule is x_next=(x+S/x)/2. Its input values are Target S, Current guess x, and the main result is Next approximation. For example: For S=10 and guess 3, the next guess is 3.1667.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated newton square-root step relation over the valid real-number domain stated below. The implemented relation is x_next=(x+S/x)/2, evaluated from Target S, Current guess x to produce Next approximation. Newton's method replaces the current guess by the tangent line's root. A zero starting guess causes division by zero; multiple steps may be needed.
Inputs and valid domain
- Target S must be a finite real number, at least 0.
- Current guess x must be a finite real number.
Important boundary: A zero starting guess causes division by zero; multiple steps may be needed.
The formula
x_next=(x+S/x)/2
How the calculator works through it
It substitutes Target S, Current guess x into the formula and exposes every numerical step above. The main output is Next approximation, accompanied by Current squared error, Next squared error.
Read the result correctly
The Next approximation is the direct answer to “perform one newton iteration toward the square root of a positive number.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
For S=10 and guess 3, the next guess is 3.1667.
Where this model stops being reliable
A zero starting guess causes division by zero; multiple steps may be needed.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Newton square-root step works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Newton square-root step uses x_next=(x+S/x)/2. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Derivatives as rates of change
Rates of change explain the local behaviour captured or approximated by Newton square-root step.
Review this foundation about 7 min
Optional enrichment
- Accumulation and integral notation
Integral notation connects Newton square-root step to accumulated change, area and continuous totals.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Target S, Current guess x.
- Evaluate the principal relationship: x_next=(x+S/x)/2.
- Return Next approximation and check the domain conditions described above.
Python
from math import *
def newton_square_root_step(a, x) -> float:
return ((x + (a / x)) / 2.0)
assert abs(newton_square_root_step(10, 3) - 3.166666666666667) < 1e-6 * max(1.0, abs(3.166666666666667))
C
#include <assert.h>
#include <math.h>
double newton_square_root_step(double a, double x) {
return ((x + (a / x)) / 2.0);
}
int main(void) {
const double expected = 3.166666666666667;
const double actual = newton_square_root_step(10, 3);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double newton_square_root_step(double a, double x) {
return ((x + (a / x)) / 2.0);
}
int main() {
constexpr double expected = 3.166666666666667;
const double actual = newton_square_root_step(10, 3);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double newton_square_root_step(double a, double x)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global newton_square_root_step
section .text
newton_square_root_step:
push rbp
mov rbp, rsp
sub rsp, 48
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd xmm0, [rbp-8]
divsd xmm0, [rbp-16]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-16]
addsd xmm0, [rbp-40]
movsd [rbp-32], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-32]
divsd xmm0, [rbp-48]
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = newton_square_root_step(a, x)
result = ((x + (a / x)) / 2.0);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, x_] := ((x + (a / x)) / 2.0);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Calculus Volume 1
Read OpenStax Calculus: Derivatives and integrationCite this book
- APA 7
- Strang, G., & Herman, E. (2016). Calculus volume 1. OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction
- MLA 9
- Strang, Gilbert, and Edwin Herman. Calculus Volume 1. OpenStax, 2016, https://openstax.org/books/calculus-volume-1/pages/1-introduction.
- Chicago author-date
- Strang, Gilbert, and Edwin Herman. 2016. Calculus Volume 1. Houston, TX: OpenStax. https://openstax.org/books/calculus-volume-1/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Newton Square-Root Iteration Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/newton-square-root-iteration
MLA 9
MW SysArc. “Newton Square-Root Iteration Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/newton-square-root-iteration. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Newton Square-Root Iteration Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/calculus/newton-square-root-iteration.
Harvard
MW SysArc (2026) ‘Newton Square-Root Iteration Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/newton-square-root-iteration (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_newton_square_root_step_2026,
author = {{MW SysArc}},
title = {Newton Square-Root Iteration Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/calculus/newton-square-root-iteration},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Newton Square-Root Iteration Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/calculus/newton-square-root-iteration
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Newton square-root step do?
Perform one Newton iteration toward the square root of a positive number.
How does the Newton square-root step work?
The calculator applies x_next=(x+S/x)/2. Newton's method replaces the current guess by the tangent line's root.
What can I learn from the Newton square-root step?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .