DEV Community: Caleb Zhao

Incorrect calculations: tand(x) and cotd(x)

Caleb Zhao — Fri, 15 Nov 2024 00:30:02 +0000

In this blog post, we explore incorrect calculations of the tangent function tand(x) and cotangent function cotd(x) of which the independent variable is in degrees in MATLAB.

Example 1. Given that x1 = 444555666.777 and x2=444555666.777e16, calculate tand(x1), cotd(x1), tand(x2), and cotd(x2).

Using MATLAB, I computed the values of tand(x) and cosd(x) at x1 and x2. The results are shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are $-1.337846899\red{312778}, -0.747469684\red{695369}, -\red{9.514364454222584},$ and $-0.\red{105104235265676}$ respectively.

However, the correct values, rounded to 16 significant digits, are -0.1337846899800357e1, -0.7474696844229537, -0.1732050807568877e1, and -0.5773502691896258, respectively (as provided by ISRealsoft).

Therefore, the error rates for the significant digits of the first two are 6/16 = 37.5% and 7/16 = 43.75%, respectively; while the last two have completely incorrect digits, resulting in an error rate of 100% for each.

Incorrect calculations: sind(x) and cosd(x)

Caleb Zhao — Thu, 14 Nov 2024 15:36:26 +0000

In this blog post, we explore incorrect calculations of the sine function sind(x) and cosine function cosd(x) of which the independent variable is in degrees in MATLAB.

Example 1. Given that x1 = 12345678.9 and x2=12345678.9e20, calculate sind(x1), cosd(x1), sind(x2), and cosd(x2).

Using MATLAB, I computed the values of sind(x) and cosd(x) at x1 and x2. The results are shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are $-0.323917418\red{204301}, -0.94608535882\red{5439}, 0.\red{927183854566787},$ and $-0.\red{374606593415912}$ respectively.

However, the correct values for the first two, rounded to 16 significant digits, are -0.3239174181981494 and -0.9460853588275453, respectively; the accurate values for the last two are 0 and 1, respectively (as provided by ISRealsoft).

Therefore, the error rates for the significant digits of the first two are 7/16 = 43.75% and 5/16 = 31.25%, respectively; while the last two have completely incorrect digits, resulting in an error rate of 100% for each.

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Incorrect calculations: csc(x) near x=k*π

Caleb Zhao — Tue, 12 Nov 2024 02:20:34 +0000

In this blog post, we explore incorrect calculations of the cosecant function csc(x) in MATLAB near numbers around $k\,\pi.$

Example 1. Given that $x_1=0.942477796123\textup{e}1 \approx 3\,\pi$ and
$,x_2=0.7853981633974\textup{e}2 \approx 25\,\pi\,,$ calculate $csc(x_1)$ and $csc(x_2)$ in MATLAB.

Using MATLAB, we computed the values of $csc(x_1)$ and $csc(x_2).$ The result is shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are
$−2.170988725424315e+09-2.17098\red{8725424315}\textup{e}+09$ and $2.069\red{879330499836}\textup{e}+11$ , respectively.

However, the correct values with 16 significant digits are -0.217098558923038e10 and 0.2069981278717736e12, respectively (as provided by ISRealsoft).

Thus, the output of MATLAB contains only 6 and 4 correct digits, with error rates of (16-6)/16 = 62.5% and (16-4)/16 = 75% respectively for the significant figures.

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Incorrect calculations: sec(x) near x=k*π+π/2

Caleb Zhao — Mon, 11 Nov 2024 14:54:03 +0000

In this blog post, we explore incorrect calculations of the secant function sec(x) in MATLAB near numbers around $k\,\pi+\frac{\pi}{2}.$

Example 1. Given that $x_1=0.1099557429\textup{e}2 \approx 3\,\pi+\frac{\pi}{2}$ and
$,x_2=0.8011061266654\textup{e}2 \approx 25\,\pi+\frac{\pi}{2}\,,$ calculate $sec(x_1)$ and $sec(x_2)$ in MATLAB.

Using MATLAB, we computed the values of $sec(x_1)$ and $sec(x_2).$ The result is shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are
$4.105555334642597e+084.10555\red{5334642597}\textup{e}+08$ and $3.580150224915013e+123.\red{580150224915013}\textup{e}+12$ , respectively.

However, the correct values with 16 significant digits are 0.4105556037464873e9 and 0.3670813182326778e13, respectively (as provided by ISRealsoft).

Thus, the output of MATLAB contains only 6 and 1 correct digits, with error rates of (16-6)/16 = 62.5% and (16-1)/16 = 93.75% respectively for the significant figures.

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Incorrect calculations: sec(x) and csc(x) for large values of x

Caleb Zhao — Mon, 11 Nov 2024 02:55:43 +0000

In this blog post, we explore incorrect calculations of the secant function sec(x) and cosecant function csc(x) for large numbers in MATLAB.

Example 1. Given that x = 1234567890888777.123, calculate sec(x) and csc(x).

Using MATLAB, I computed the values of sec(x) and csc(x) at x = 1234567890888777.123. The result is shown in the following screenshot.

From the above screenshot, it can be seen that the outputs from MATLAB are $-1.\red{059606794245841}$ and $-\red{3.024159505568874}$ respectively.

However, the correct values with 16 significant digits are -0.1116013690832344e1 and -0.2252452617594792e1, respectively (as provided by ISRealsoft).

Therefore, out of the 32 significant digits in the two outputs from MATLAB, only 1 digit is correct. The error rates are 15/16 = 93.75% and 16/16 = 100%, respectively.

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Incorrect calculations: Horner polynomials

Caleb Zhao — Sun, 10 Nov 2024 12:08:05 +0000

In this blog post, we explore issues related to incorrect polynomial evaluations using the Horner scheme (also known as the Qin Jiushao algorithm) within MATLAB.

Example 1. Consider the polynomial function f(x) defined as:

f(x) = ((((((c7*x+c6)*x+c5)*x+c4)*x+c3)*x+c2)*x+c1)*x+c0,

where the coefficients are given by c7 = 111990.0, c6 = 5115500000.0, c5 = 949100000.0, c4 = 4058000000.0, c3 = 1531000000.0, c2 = 5781000000.0, c1 = 350800000.0, and c0 = -321499872.52. Calculate the value of f(x) at x = 45678.0.

Using MATLAB, I computed the value of the polynomial at x = 45678.0. The result is shown in the following screenshot.

From the above screenshot, it can be seen that the output from MATLAB is a large number having 38 integer digits.

However, the accurate value of the polynomial is -0.64307299252e9 (as provided by ISRealsoft). Thus, the output from MATLAB is incorrect.

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Incorrect calculations: cos(x) near x=k*π+π/2

Caleb Zhao — Fri, 08 Nov 2024 08:30:00 +0000

In this blog post, we explore incorrect calculations of the cosine function cos(x) in MATLAB near numbers around $k\,\pi+\frac{\pi}{2}.$

Example 1. Given that $x_1=0.392699082\textup{e}2\approx 12\,\pi+\frac{\pi}{2}$ and
$,x_2=0.1115265392024\textup{e}3\approx 35\,\pi+\frac{\pi}{2}\,,$ calculate $cos(x_1)$ and $cos(x_2)$ in MATLAB.

Let's just post the figure directly.

From the above screenshot, it can be seen that the two outputs from MATLAB are
$-3.012758\red{778139255}e-08$ and $-3.765\red{778342090408}e-11.$

However, the correct values with 16 significant digits are -0.3012758451921695e-7 and -0.3765996542384011e-10, respectively (as provided by ISRealsoft).

Thus, the outputs of MATLAB contain 9 and 12 incorrect digits, with error rates of 9/16 = 56.25% and 12/16 = 75% respectively for the significant figures.

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Incorrect calculations: sin(x) near x=k*π

Caleb Zhao — Fri, 08 Nov 2024 02:33:47 +0000

In this blog post, we explore incorrect calculations of the sine function sin(x) in MATLAB near numbers around $k\,\pi (k\neq0).$

Example 1. Given that $x_1=0.3769911184\textup{e}2\approx 12\,\pi$ and
$,x_2=0.166504\textup{e}3\approx 53\,\pi\,,$ calculate $sin(x_1)$ and $sin(x_2)$ in MATLAB.

Let's just post the figure directly.

From the above screenshot, it can be seen that the two outputs from MATLAB are
$-3.077518\red{377555020}e-09$ and $4.106402475\red{102297}e-04.$

However, the correct values with 16 significant digits are -0.3077518861551721e-8 and 0.4106402475009074e-3, respectively (as provided by ISRealsoft).

Thus, the outputs of MATLAB contain 9 and 6 incorrect digits, with error rates of 9/16 = 56.25% and 6/16 = 37.5% respectively for the significant figures.

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Incorrect calculations: polynomials

Caleb Zhao — Wed, 06 Nov 2024 15:24:47 +0000

In this blog post, we discuss incorrect calculations of polynomials in MATLAB.

Example 1. Given that

f(x)=727313\,x^{12}-89459499\,x^{11}-44\,x^2+665683.13

and

g(x)=584927\,x^9-721800000\,x^8+101188\,x^7-13\,x^2+19795832\,,

calculate

f (123)

and

.g(1234)\,.

Let's just post the figure directly.

From the above screenshot, it can be seen that the two outputs from MATLAB are both large numbers, with one having 16 integer digits and the other having 17 integer digits.

However, the accurate values of the two polynomials are 7.13 and 4, respectively (as provided by ISRealsoft). Thus, the outputs from MATLAB are both incorrect.

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Incorrect calculations: tan(x) and cot(x) for large values of x

Caleb Zhao — Tue, 05 Nov 2024 08:22:01 +0000

In this blog post, we explore incorrect calculations of the tangent function tan(x) and the cotangent function cot(x) for large numbers in MATLAB.

Example 1. Suppose $x=235556677777.87\,.$ Calculate $\tan(x)$ and $\cot(x)$ in MATLAB.

Let's just post the figure directly.

From the above figure, it can be seen that the two outputs from MATLAB are
$1.4290\red{64236906033}$ and $0.69975\red{8607188316} .$

However, the correct values with 16 significant digits are 0.1429079091621959e1 and 0.6997513334724058, respectively (as provided by ISRealsoft).

As a result, both outputs from MATLAB contain 11 incorrect digits, with an error rate of 11/16 = 68.75% for the significant figures.

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Incorrect calculations: tan(x) and cot(x) near x=k*π

Caleb Zhao — Mon, 04 Nov 2024 08:56:31 +0000

Exploring the computational errors of the tangent function tan(x) and cotangent function cot(x) in MATLAB near numbers around $k\,\pi (k\neq0).$

Example 1. Suppose $x=65.973446\approx21\,\pi.$ Calculate $\tan(x)$ and $\cot(x)$ in MATLAB.

Let's just post the figure directly.

From the above figure, it can be seen that the two outputs from MATLAB are
$2.746143\red{375848801}e-07$ and $3.6414704\red{66526212}e+06.$
However, the correct values with 16 significant digits are 0.2746143419922914e-6 and 0.3641470408082585e7, respectively (as provided by ISRealsoft).
Thus, the outputs of MATLAB contain 9 and 8 incorrect digits, with error rates of 9/16 = 56.25% and 8/16 = 50% respectively for the significant figures.

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Incorrect calculations: tan(x) and cot(x) near x=k*π+π/2

Caleb Zhao — Sun, 03 Nov 2024 13:29:16 +0000

Exploring the computational errors of the tangent function tan(x) and cotangent function cot(x) in MATLAB near numbers around $k\,\pi+\frac{\pi}{2}.$

Example 1. Suppose $x=39.26990817\approx12\,\pi+\frac{\pi}{2}.$ Calculate $\tan(x)$ and $\cot(x)$ in MATLAB.

Let's just post the picture directly.

However, the correct values with 16 significant digits are -0.7837941516239115e10 and -0.1275845192169577e-9, respectively (as provided by ISRealsoft).
Thus, the output of MATLAB contains only 4 and 5 correct digits, with error rates of (16-4)/16 = 75% and (16-5)/16 = 68.75% respectively for the significant figures.

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