Mathematics · Calculus

Armijo Sufficient-Decrease Threshold Calculator

Calculate required decrease magnitude from step length and scaled descent-slope magnitude.

Runs locally
Your numbers

Inputs and results stay in this browser. Change one value at a time to explore the relationship.

Your inputCalculatedPassed forward in chains
required decrease magnitude0.003

Calculation steps

  1. Use c=ab with step length=0.2 and scaled descent-slope magnitude=0.015.
  2. required decrease magnitude=0.003.

Understand Armijo Sufficient-Decrease Threshold

One idea, three depths

Choose how deeply to explain Armijo Sufficient-Decrease Threshold

Armijo Sufficient-Decrease Threshold: Calculate required decrease magnitude from step length and scaled descent-slope magnitude.

Age 5Explain it to a 5-year-oldStart with a picture

Imagine using Armijo Sufficient-Decrease Threshold to answer this question: calculate required decrease magnitude from step length and scaled descent-slope magnitude? Enter step length and scaled descent-slope magnitude; the calculator shows required decrease magnitude. For example: step length=0.2 and scaled descent-slope magnitude=0.015 produce required decrease magnitude=0.003. The answer tells you required decrease magnitude.

Age 15Explain it to a 15-year-oldConnect it to the formula

The Armijo decrease threshold multiplies step length by the descent slope magnitude after applying the Armijo coefficient. This page evaluates the relationship directly. The rule is c=ab. Its input values are step length, scaled descent-slope magnitude, and the main result is required decrease magnitude. For example: step length=0.2 and scaled descent-slope magnitude=0.015 produce required decrease magnitude=0.003.

CollegeExplain it at college levelState the model precisely

This calculator evaluates the stated armijo sufficient-decrease threshold relation over the valid real-number domain stated below. The implemented relation is c=ab, evaluated from step length, scaled descent-slope magnitude to produce required decrease magnitude. The Armijo decrease threshold multiplies step length by the descent slope magnitude after applying the Armijo coefficient. This page evaluates the relationship directly. Enter a positive magnitude after handling the directional derivative sign.

Inputs and valid domain

  • step length must be a finite real number.
  • scaled descent-slope magnitude must be a finite real number.

Important boundary: Enter a positive magnitude after handling the directional derivative sign.

The formula

c=ab

How the calculator works through it

It substitutes step length, scaled descent-slope magnitude into the formula and exposes every numerical step above. The main output is required decrease magnitude.

Read the result correctly

The required decrease magnitude is the direct answer to “calculate required decrease magnitude from step length and scaled descent-slope magnitude.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.

A worked check

step length=0.2 and scaled descent-slope magnitude=0.015 produce required decrease magnitude=0.003.

Where this model stops being reliable

Enter a positive magnitude after handling the directional derivative sign.

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 Armijo Sufficient-Decrease Threshold works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.

Hard requirements

  • Reading formulas and substituting values

    Armijo Sufficient-Decrease Threshold uses c=ab. 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

Optional enrichment

  • Accumulation and integral notation

    Integral notation connects Armijo Sufficient-Decrease Threshold to accumulated change, area and continuous totals.

    Review this foundation about 6 min
Learn the missing foundationsI already know these — show the code

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

  1. Read step length, scaled descent-slope magnitude.
  2. Evaluate the principal relationship: c=ab.
  3. Return required decrease magnitude and check the domain conditions described above.
Python
            from math import *

def armijo_decrease_threshold_calculator(a, b) -> float:
    return (a * b)

assert abs(armijo_decrease_threshold_calculator(0.2, 0.015) - 0.003) < 1e-6 * max(1.0, abs(0.003))
          
Current calculator valuesUpdates when you change an input above.
              
            
C
            #include <assert.h>
#include <math.h>

double armijo_decrease_threshold_calculator(double a, double b) {
    return (a * b);
}

int main(void) {
    const double expected = 0.003;
    const double actual = armijo_decrease_threshold_calculator(0.2, 0.015);
    assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
C++
            #include <cassert>
#include <cmath>
#include <numbers>

double armijo_decrease_threshold_calculator(double a, double b) {
    return (a * b);
}

int main() {
    constexpr double expected = 0.003;
    const double actual = armijo_decrease_threshold_calculator(0.2, 0.015);
    assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
          
Current calculator valuesUpdates when you change an input above.
              
            
Linux x86-64 assembly

x86-64 NASM · System V ABI · Linux · SSE2 with libm where required

            ; double armijo_decrease_threshold_calculator(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global armijo_decrease_threshold_calculator
section .text

armijo_decrease_threshold_calculator:
    push rbp
    mov rbp, rsp
    sub rsp, 32
    movsd [rbp-8], xmm0
    movsd [rbp-16], xmm1
    movsd xmm0, [rbp-8]
    mulsd xmm0, [rbp-16]
    movsd [rbp-24], xmm0
    movsd xmm0, [rbp-24]
    leave
    ret
          
Current calculator valuesUpdates when you change an input above.
              
            
MATLAB
            function result = armijo_decrease_threshold_calculator(a, b)
    result = (a * b);
end
          
Current calculator valuesUpdates when you change an input above.
              
            
Wolfram Language
            ClearAll[mwCalculate];
mwCalculate[a_, b_] := (a * b);
          
Current calculator valuesUpdates when you change an input above.
              
            

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 integration
Cite 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). Armijo Sufficient-Decrease Threshold Calculator. MW SysArc Tools. https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator

MLA 9

MW SysArc. “Armijo Sufficient-Decrease Threshold Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator. Accessed 31 Aug. 2026.

Chicago 17

MW SysArc. “Armijo Sufficient-Decrease Threshold Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator.

Harvard

MW SysArc (2026) ‘Armijo Sufficient-Decrease Threshold Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator (Accessed: 31 August 2026).

BibTeX and RIS records

BibTeX

@misc{mwsysarc_armijo_decrease_threshold_calculator_2026,
  author = {{MW SysArc}},
  title = {Armijo Sufficient-Decrease Threshold Calculator},
  howpublished = {MW SysArc Tools},
  year = {2026},
  url = {https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator},
  note = {Published July 21, 2026; accessed August 31, 2026}
}

RIS

TY  - ELEC
AU  - MW SysArc
TI  - Armijo Sufficient-Decrease Threshold Calculator
T2  - MW SysArc Tools
PY  - 2026
DA  - 2026-07-21
Y2  - 2026-08-31
UR  - https://math.mwsysarc.com/calculus/armijo-decrease-threshold-calculator
N1  - Published July 21, 2026
ER  -

Clear answers

Frequently asked questions

What does the Armijo Sufficient-Decrease Threshold do?

Calculate required decrease magnitude from step length and scaled descent-slope magnitude.

How does the Armijo Sufficient-Decrease Threshold work?

The calculator applies c=ab. The Armijo decrease threshold multiplies step length by the descent slope magnitude after applying the Armijo coefficient. This page evaluates the relationship directly.

What can I learn from the Armijo Sufficient-Decrease Threshold?

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 .

MW SysArc Certified