text:
Medical Care Technologies Inc. Announces Financing Approval For Medical Equipment
BEIJING, Feb. 4, 2013 /PRNewswire/ -- Medical Care Technologies Inc. , a growing American medical company providing information technology systems, family and children's healthcare services and Western healthcare products for distribution in China,
BEIJING, Feb. 4, 2013 /PRNewswire/ -- Medical Care Technologies Inc. , a growing American medical company providing information technology systems, family and children's healthcare services and Western healthcare products for distribution in China, is pleased to announce that it has received approval from a major Australian financing firm who will underwrite a minimum of US $500,000 in equipment financing to fund the Company's acquisition of medical equipment for its Shenzhen health clinic.
The financing will be guaranteed by a principal of the Company's Chinese joint venture partner and secured by the medical equipment purchased. Thus far, MDCE has identified and sourced pricing of US $365,000 in medical equipment and clinical tools for its Shenzhen clinic. Terms of the financing are expected to be in the range of 36-60 months, with an effective interest rate of between 6-8% and can be repaid without penalty at any time. Title to the medical equipment will transfer to the Shenzhen clinic operations at the completion of the term. The financing proposal is subject to customary closing conditions.
Medical Care Technologies Inc. announced last fall, the signing of an MOU with Dragon Link Group Ltd. of Hong Kong, for an infrastructure financing of approximately US $1.5M. With a minimum US $500,000 approval from the Australian underwriter, Dragon Link's infrastructure financing will reduce to US $1M. This arrangement between Dragon Link and MDCE, which encompasses the Shenzhen clinic, a Dongguan location, and future operations across Guangdong Province, is still very much in play. Dragon Link's extensive due diligence of MDCE's proposed clinic operations is nearing completion and expects to close definitive documents in due time.
"Securing this important element of our equipment financing is essential to us meeting our healthcare services objectives," commented Luis Kuo, President. "The financial support of this major underwriter further validates our belief that the Shenzhen clinic is an excellent opportunity with significant potential."
"Our unwavering goal is to lay an unshakeable foundation for our Western-style healthcare initiatives across China, initiatives that will prosper for decades and will become cornerstones of China's future healthcare system", said CEO Ning Wu. "We are taking the time to ensure that every step we take in these crucial days of our clinic's development stage is measured and sure-footed," she added.
About Medical Care Technologies Inc.
Medical Care Technologies Inc. is traded under the symbol MDCE on the OTCBB and is headquartered in Beijing, China. MDCE, through joint ventures or Chinese subsidiaries, develops a network of children's health facilities in the larger urban areas throughout China. Services are geared towards the advancing economic middle-class Chinese families. Specializing in the care of children between the ages of 3 to 16, MDCE's role is to enhance the overall well-being of the family and community and to expand its pediatric services to include preventative health and wellness education. MDCE, through its children's health facilities, will also distribute a diverse range of industry-leading pharmaceutical and nutraceutical product lines. MDCE's main mission is simple - to become a healthcare service provider leader in children's health. Information on the Company can be found at www.sec.gov and the Company's website at www.medicaretechinc.com.
Safe Harbor Statement
All statements contained in this press release, other than statements of historical fact, are forward-looking statements, including those regarding: MDCE's products, services, capabilities, performance, opportunities, development and business outlook, guidance on our future financial results and other projections or measures of our future performance; the amount and timing of the benefits expected from strategic initiatives and acquisitions or from deployment of new or updated technologies, products, services or applications; and other potential sources of additional revenue. These statements are based on our current plans and expectations and involve risks and uncertainties that could cause actual future events or results to be different than those described in or implied by such forward-looking statements. These risks and uncertainties include those relating to: lack of operating history, transitioning from a development company to an operating company, difficulties in distinguishing MDCE's products and services, ability to deploy MDCE's services and products, market acceptance of our products and services; operational difficulties relating to combining acquired companies and businesses; our ability to form and maintain mutually beneficial relationships with customers and strategic partners; changes in economic, political or regulatory conditions or other trends affecting the healthcare, Internet, information technology and healthcare and pharmaceutical industries, and our ability to attract and retain qualified personnel. Other risks and uncertainties may include, but are not limited to: lack of or delay in market acceptance and fluctuations in customer demand, dependence on a limited number of significant customers, reliance on third party vendors and strategic partners, ability to meet future capital requirements on acceptable terms, continuing uncertainty in the global economy, and compliance with federal and state regulatory requirement. Further information about these matters can be found in our Securities and Exchange Commission filings. We expressly disclaim any intent or obligation to update these forward-looking statements.
For Further Information:
Contact: Mr. Peter Verner
Email: firstname.lastname@example.com
Tel: (852) 8122-9660
Web: www.medicaretechinc.com


text:
Blue Double Sided Floral Italian Silk Pocket Square
Made in Italy by Tessitura Attilio Bottinelli. Como has been the centre of the Italian silk industry since the 15th century and the Bottinelli family have been weavers there since the 1930s. Still very much a family business, we work every season with Roberto & Paolo, grandsons of the founder, drawing on their technical knowledge to realise our designs. We believe the benefits of this close collaboration is what gives our pocket squares an edge, and you too should you choose to wear one.
- 100% Silk
- Made in Italy
- Dry Clean Only
More pocket squares
Read more about shipping on our Shipping page.
Check out the full returns process on our Returns page.

subset: commoncrawl


text: To cook and enjoy as a main course. Traditional cooking: brown the content of the packet in a tablespoonful of oil. Progressively add 650 ml to 750 ml of water while stirring. Let the mixture boil for about 15 to 17 min. Pressure cooking: pour the packet into half a liter of cold water. Bring up to high heat and cook for 8 min after the hissing noise starts. Suggestions: for a classic preparation, add 1/3 of a cup of white wine before adding water. For a creamier result, melt in some butter or parmesan cheese before serving.


text:
Two new low cost handsets are now available for your budget-minded buying pleasure. This weekend AT&T debuted its teen and tween texting machine, the Motorola Karma QA1. The quad-band EDGE/dual-band HSDPA handset features a sliding QWERTY keyboard, 2.5-inch QVGA display, 2 megapixel camera, aGPS, Bluetooth 2.0 with A2DP, 3.5mm headphone jack, microSDHC expansion and, most important for socializing t(w)eens, Facebook, MySpace, IM, MMS and threaded SMS support. The Karma is available now for $79.99 with a $50 mail in rebate and a 2-year contract.
On the CDMA side of things, we find the HTC Ozone. The Ozone is a Windows Mobile 6.1 Standard handset with dual-band 1xRTT/EV-DO Rev. A, quad-band GSM/GPRS/EDGE for international roaming, 2.4-inch QVGA display, QWERTY keyboard with 5-way navigation key, 2 megapixel camera w/ auto-focus and video capture, 256MB ROM/128MB RAM, Wi-Fi 802.11b/g, Bluetooth 2.0 with A2DP and microSD expansion (up to 16GB). The Ozone is available now for $49.99 after $70 mail-in rebate and 2-year contract via Web, telesales and B2B channels with in-store availability expected on July 13th.


text:
Year after NH hepatitis C outbreak, testing slows
Tainted syringes left 6 in Kansas with hepatitis C
CONCORD, N.H. — The former employers of a New Hampshire medical technician charged a year ago in a hepatitis C outbreak are still testing patients he might have infected, but the pace has slowed in recent months.
Hospital and public health officials tell The Associated Press that only a few hundred patients have been tested since December, and at least 3,300 of the nearly 7,900 people asked to get tested outside of New Hampshire have yet to do so.
Altogether, 47 people have been diagnosed with the same strain of hepatitis C that David Kwiatkowski carries and is accused of spreading through tainted syringes — 33 in New Hampshire, seven in Maryland, six in Kansas and one on Pennsylvania.
Kwiatkowski has pleaded not guilty to federal drug charges.


text:
1 in 4 men don't think exposing themselves constitutes sexual harassment, survey finds
Perhaps not surprisingly, men and women are not on the same page when it comes to what constitutes sexual harassment, a survey by the Barna Group found.
The research group asked over 1,000 Americans "What do you think counts as sexual harassment?" in an online survey. They were told to select all that apply.
The survey found that 91 percent of women and 83 percent of men thought "being forced to do something sexual" fit the bill.
If that doesn't count as sexual harassment, what does? Who are the 17 percent of men and 9 percent of women who don't agree?
Men were even less likely to characterize behaviors that didn't involve physical touch as sexual harassment. Twenty-four percent thought flashing someone doesn't count and 30 percent thought making sexual comments about someone's body isn't harassment.
What constitutes harassment in the workplace?
Sarah Layden, director of advocacy services at Rape Victim Advocates, joins the set to talk about defining harassment in the workplace.
Nearly half of men thought pushing up against someone on public transportation doesn't constitute harassment, but most women (70 percent) disagreed.
Merriam-Webster defines sexual harassment as "uninvited and unwelcome verbal or physical behavior of a sexual nature especially by a person in authority toward a subordinate (such as an employee or student)."
Women were consistently more likely to consider a behavior a sexual harassment in all but one category of the Barna Group survey.
The research group didn't provide detailed explanations on why its respondents may have chosen as they did, but it's possible to imagine a few complications with the survey's wording. For example, some may have thought behaviors like following someone or staring should be defined as harassment, not necessarily sexual harassment.
It's also possible that in some cases, respondents thought the categorization of "sexual harassment" was not harsh enough, and that some behavior should be identified as "sexual assault."
It's not clear whether respondents were provided with a definition of sexual harassment before participating in the survey.
You can read the full results of the survey here.


text:
Story highlights
Paris food blogger David Lebovitz goes on a quest for the city's best croissant
Au Levain du Marais provides extra-crisp ends
In Montmartre, Gontran Cherrier's croissants are resolutely unadorned
Pâtisserie Boulangerie Blé Sucré's croissant is set apart by a hint of sea salt
I'm not an early riser. I have a rule against stepping outside my Paris apartment without first indulging in my café au lait.
But after close to a decade of living here, I also know that if you're craving a croissant, you have to go to a bakery well before noon to taste the city's iconic breakfast staple at its most exquisite: still slightly warm from the oven, the irresistible aroma of freshly baked dough wafting upward while you're pulling the whole thing apart, layer by buttery layer.
So on a recent morning, I rose early and set out to find the city's top pastry.
Others have gone on a quest for the perfect croissant, but always landed on swanky Left Bank addresses. I thought it was time to give the Right Bank its due, now that a new generation of young bakers have set up their ovens.
Travel + Leisure: Affordable small hotels in Paris
To start, I didn't have to go far from my apartment in the lively Bastille quarter. In fact, I went right next door to Au Levain du Marais. I've long appreciated this bakery's croissant for its extra-crisp ends—the best part—while the delicately layered center was soft and tender. It was a promising first stop, but there was more tasting to be done.
Heading up the hill toward Montmartre, where shaggy-haired Gontran Cherrier is known for spicing up baguettes with curry and rye bread with miso, I was surprised to find his croissant resolutely unadorned. The distinct layers, more shards than little flakes, had more of a crunch than the dainty bite I had hoped for.
Travel + Leisure: Europe's best neighborhoods for foodies
An equally obsessed friend sent me to a busy corner of Rue Paul Bert for La Pâtisserie by Cyril Lignac, whose owner is a spirited presence on French television. Here, the horseshoe-shaped croissants were lined up right next to the register. I ripped off one crisp end and let the wisps of warm, yeasted puff pastry melt in my mouth.
This one stopped me in my tracks. Was this the end of my quest?
Never one to shy away from being thorough, I made a beeline for Du Pain et des Idées, in the scenic Canal St.-Martin neighborhood. Christophe Vasseur's rustic breads are among the best in Paris, and I was curious to see how his artisanal croissant stacked up.
This specimen was more substantial than the others, with a sturdy crust and hearty interior—too dense for my Proustian ideal. But I couldn't resist a crusty loaf of my favorite pain aux céréales for later.
Travel + Leisure: World's strangest desserts
When I arrived in the charming Square Trousseau, I was worried I wouldn't have room for the hefty croissant at Pâtisserie Boulangerie Blé Sucré. One look at its crackly glazed crust, one bite into the tender interior, and I was enchanted.
What set it apart? A masterful hint of French sea salt in the dough. Satisfied that I had found the ne plus ultra of croissants, I strolled out of the bakery, vowing to return the next morning, leaving a trail of crumbs in my wake.
Paris Croissants Address Book
Au Levain du Marais - 28 Blvd. Beaumarchais, 11th Arr.; croissants for two $2.50.
Du Pain et des Idées - 34 Rue Yves Toudic, 10th Arr.; croissants for two $3.
Gontran Cherrier - 22 Rue Caulaincourt, 18th Arr.; croissants for two $2.50.
La Pâtisserie by Cyril Lignac - 24 Rue Paul Bert, 11th Arr.; croissants for two $2.50.
Pâtisserie Boulangerie Blé Sucré - 7 Rue Antoine Vollon, 12th Arr.; croissants for two $2.50.
David Lebovitz is a former pastry chef and the author of "The Sweet Life in Paris" (Broadway). Follow him on Twitter @davidlebovitz.
Planning a getaway? Don't miss Travel + Leisure's guide to the World's Best Hotels
Copyright 2015 American Express Publishing Corporation. All rights reserved.


text:
My birthday is coming up later this week. I have what I would call a love hate relationship with birthdays, I hate getting older but looove getting presents. This year I'm feeling a little neutral about it though. I guess compared to the
stress/panic excitement of 30, 31 is just no big deal. Besides, one of my coworkers told me I look 27 the other day and I'm choosing to believe her.
Despite my neutral feelings, I will never pass up an opportunity for a treat. I still have a ton of sweets left over from Christmas so I decided to follow Sarah of The House Wife Diary's lead and create something out of them.
For Christmas I made Anna Olson's Marzipan Cake. This cake was incredible, it was moist and tender with a lovely almond flavour. I replaced the lemon zest with orange, which went very well with the almond. I decided to use the leftover cake to make myself a mini trifle. Then I started thinking about all the sugar I've been eating and thought maybe I should tone it down. Traditionally a trifle has layers of cake, fruit, custard, and cream. I replaced the custard and whipped cream with greek yogurt. I left the yogurt unsweetened as the cake and jam I used were both quite sweet. If that's too tangy for you, try sweetening it with a little honey.
Marzipan Cake
- 1 cup unsalted butter, room temperature
- 1/2 cup sugar
- 1 cup marzipan
- 6 eggs
- 2 teaspoons orange zest
- 1 teaspoon vanilla extract
- 1/2 teaspoon almond extract
- 1 cup all purpose flour
- 3/4 teaspoon baking powder
- 1/2 teaspoon salt
- 6 tablespoons icing sugar, sifted, for dusting
- Preheat the oven to 350. Grease an 8 inch round pan and line with parchment paper.
- In a large bowl, cream together the butter and sugar. Cut the marzipan into chunks and beat into the butter mixture one piece at a time until its completely incorporated.
- Add the eggs one at a time, mixing well after each addition. Mix in the zest and extracts.
- In separate bowl sift together the flour, baking powder and salt. Add to the marzipan mixture, beating until well incorporated.
- Pour batter into the prepared pan. Bake for 35 to 45 minutes.
Greek Yogurt Trifle
- Leftover marzipan cake
- Greek yogurt
- Raspberry jam
- Layer the cake, jam and yogurt into a tall glass or bowl. Let chill until ready to serve.


text:
Burke, VA' PREMIER HOUSE WASHING SERVICE
House washing is necessary to ensure your home is looking its best at all times
House Washing Services
There are many benefits to house washing, including improved curb appeal, removing dirt and mold, and protecting your home from potential damage. House washing can also help extend the life of your paint job and protect your investment. Whether you sell your home or want to keep it looking its best, house washing is a great option. House washing can be an effective way to clean the exterior of your home.
House washing can also help extend the life of your siding and paint by protecting it from the elements. When done correctly, pressure washing is safe for your house! Our house washing professionals take great care to deliver a safe, efficient, and thoroughly clean. We do this by employing a low to no-pressure "soft washing" technique that utilizes gently pressured water and eco-friendly cleaning solutions to erase dirt and grime from the surface of your house.
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House washing improves the curb appeal and beauty of your home. It can make your home look cleaner, but it can also help remove dirt, mold, and algae.
Roof cleaning is also beneficial for maintaining the value of your home but can also help extend its lifespan and prevent further damage from occurring.
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Cleaning Tips & Articles
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When clients field suggestions for companies to purchase goods or services from, there's one word that comes up more often than most: "trustworthy". What, exactly, does "trustworthy" mean in the cont […]
What Is Soft Washing?
If you're not familiar, or are only slightly familiar, with pressure washing, soft washing may sound like a foreign concept. However, it's the primary cleaning method we use to wash your property. As […]
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Today the team helped one of our valued real estate customers with the final touches of the home before putting it on the market for sale. We completed a full window cleaning package which includes in […]
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text:
Letra canción To Be a Princess - Barbie
Now remember,
No nagging, bragging, sweating, fretting, slipping, tripping,
Slurping, burping, twittering or frittering allowed.
Stay Present, stay pleasant, stay proud.
To be a princess
Is to know which spoon to use
To be a princess
Is to own a thousand pairs of shoes
To maintain a regal gait
Leave the parsley on your plate
And be charming but detached
And yet amused
To be a princess
Is to never be confused
Do a plié and never fall
Don't ever stray from protocol
All through the day
There's just one way
You must behave
Do keep a grip and never crack
Stiff upper lip and arch the back
Bend from above
And always wear your gloves
And wave
Shoulders back and
Tummy in and
Pinky out and
Lift the chin and
Slowly turn the head from side to side
I see now
Breathing gently
Stepping lightly
Smile brightly
Nod politely
Never show a thing you feel inside
To be a princess
Is to always look your best
To be a princess
Is to never get to rest
Sit for a portrait, never squirm
Sleep on a mattress extra firm
Speak and be clever
Never at a loss for words
Curtsy to every count and lord
Learn how to play the harpsichord
Sing lullabies and always harmonize in thirds
Thirds, thirds
She has beautiful eyes
Your spirits rise
When she walks in the room
I see
Doors close
And the chemistry grows
She's like a rose
That's forever in bloom
Do a plié and don't
(Ever ever ever fall)
Never show dismay
(And be there when people call)
Be prepared whatever royal life will bring
Do keep a grip and don't
(Ever ever ever crack)
Take a dainty sip
(Never ever turn your back)
There's a time and place and way for everything
To be a princess
Is to never make your bed
To be a princess
Is to always use your head
- visitas: 250
- 02-04-2021
- Canciones


text:
I'm about to head out. It occurs to me that tonight begs for a drinking game. Mark Jones is enough of a hack that we ought to hear plenty about the awful doings from last year.
One wrinkle – I'll let you guys come up with the rules. So have at it in the comments.


text:
Floating-Point Numbers
MATLAB® represents floating-point numbers in either double-precision or single-precision format. The default is double precision, but you can make any number single precision with a simple conversion function.
Double-Precision Floating Point
MATLAB constructs the double-precision (or
double) data type
according to IEEE® Standard 754 for double precision. Any value stored as a
double requires 64 bits, formatted as shown in the table
Sign (
Exponent, biased by
Single-Precision Floating Point
MATLAB constructs the single-precision (or
single) data type
according to IEEE Standard 754 for single precision. Any value stored as a
single requires 32 bits, formatted as shown in the table
Sign (
Exponent, biased by
Because MATLAB stores numbers of type
single using 32 bits, they require
less memory than numbers of type
double, which use 64 bits. However,
because they are stored with fewer bits, numbers of type
single are
represented to less precision than numbers of type
Creating Floating-Point Data
Use double-precision to store values greater than approximately 3.4 x 1038 or less than approximately -3.4 x 1038. For numbers that lie between these two limits, you can use either double- or single-precision, but single requires less memory.
Creating Double-Precision Data
Because the default numeric type for MATLAB is
double, you can create a
with a simple assignment statement:
x = 25.783;
whos function shows that MATLAB has created a 1-by-1 array of type
double for the value
you just stored in
whos x Name Size Bytes Class x 1x1 8 double
isfloat if you just want to verify that
x is a floating-point number. This function returns logical 1
true) if the input is a floating-point number, and logical 0
false) otherwise:
isfloat(x) ans = logical 1
You can convert other numeric data, characters or strings, and logical data to
double precision using the MATLAB function,
double. This example converts a signed
integer to double-precision floating point:
y = int64(-589324077574); % Create a 64-bit integer x = double(y) % Convert to double x = -5.8932e+11
Creating Single-Precision Data
Because MATLAB stores numeric data as a
double by default, you need to
use the
single conversion function to create a
single-precision number:
x = single(25.783);
whos function returns the attributes of
x in a structure. The
bytes field of this
structure shows that when
x is stored as a single, it requires just 4
bytes compared with the 8 bytes to store it as a
xAttrib = whos('x'); xAttrib.bytes ans = 4
You can convert other numeric data, characters or strings, and logical data to
single precision using the
single function. This example converts a
signed integer to single-precision floating point:
y = int64(-589324077574); % Create a 64-bit integer x = single(y) % Convert to single x = single -5.8932e+11
Arithmetic Operations on Floating-Point Numbers
This section describes which classes you can use in arithmetic operations with floating-point numbers.
Double-Precision Operations
You can perform basic arithmetic operations with
double and any
of the following other classes. When one or more operands is an integer (scalar or
array), the
double operand must be a scalar. The result is of type
double, except where noted otherwise:
single— The result is of type
uint*— The result has the same data type as the integer operand
This example performs arithmetic on data of types
char and
double. The result is of type
c = 'uppercase' - 32; class(c) ans = double char(c) ans = UPPERCASE
Single-Precision Operations
You can perform basic arithmetic operations with
single and any
of the following other classes. The result is always
In this example, 7.5 defaults to type
double, and the result is
of type
x = single([1.32 3.47 5.28]) .* 7.5; class(x) ans = single
Largest and Smallest Values for Floating-Point Classes
For the
double and
single classes, there is a
largest and smallest number that you can represent with that type.
Largest and Smallest Double-Precision Values
The MATLAB functions
realmax and
realmin return the maximum and minimum values that you can represent with
double data type:
str = 'The range for double is:\n\t%g to %g and\n\t %g to %g'; sprintf(str, -realmax, -realmin, realmin, realmax) ans = The range for double is: -1.79769e+308 to -2.22507e-308 and 2.22507e-308 to 1.79769e+308
Numbers larger than
realmax or smaller than
-realmax are assigned the values of positive and negative infinity,
realmax + .0001e+308 ans = Inf -realmax - .0001e+308 ans = -Inf
Largest and Smallest Single-Precision Values
The MATLAB functions
realmax and
realmin, when called with the argument
'single', return
the maximum and minimum values that you can represent with the
data type:
str = 'The range for single is:\n\t%g to %g and\n\t %g to %g'; sprintf(str, -realmax('single'), -realmin('single'), ... realmin('single'), realmax('single')) ans = The range for single is: -3.40282e+38 to -1.17549e-38 and 1.17549e-38 to 3.40282e+38
Numbers larger than
realmax('single') or smaller than
-realmax('single') are assigned the values of positive and negative
infinity, respectively:
realmax('single') + .0001e+038 ans = single Inf -realmax('single') - .0001e+038 ans = single -Inf
Accuracy of Floating-Point Data
If the result of a floating-point arithmetic computation is not as precise as you had expected, it is likely caused by the limitations of your computer's hardware. Probably, your result was a little less exact because the hardware had insufficient bits to represent the result with perfect accuracy; therefore, it truncated the resulting value.
Double-Precision Accuracy
Because there are only a finite number of double-precision numbers, you cannot
represent all numbers in double-precision storage. On any computer, there is a small gap
between each double-precision number and the next larger double-precision number. You
can determine the size of this gap, which limits the precision of your results, using
eps function. For example, to find the
distance between
5 and the next larger double-precision number,
format long eps(5) ans = 8.881784197001252e-16
This tells you that there are no double-precision numbers between 5 and
5 + eps(5). If a double-precision computation returns the
answer 5, the result is only accurate to within
The value of
eps(x) depends on
x. This example
shows that, as
x gets larger, so does
eps(50) ans = 7.105427357601002e-15
If you enter
eps with no input argument, MATLAB returns the value of
eps(1), the distance from
1 to the next larger double-precision number.
Single-Precision Accuracy
Similarly, there are gaps between any two single-precision numbers. If
x has type
returns the distance between
x and the next larger single-precision
number. For example,
x = single(5); eps(x)
ans = single 4.7684e-07
Note that this result is larger than
eps(5). Because there are
fewer single-precision numbers than double-precision numbers, the gaps between the
single-precision numbers are larger than the gaps between double-precision numbers. This
means that results in single-precision arithmetic are less precise than in
double-precision arithmetic.
For a number
x of type
eps(single(x)) gives you an upper bound for the amount that
x is rounded when you convert it from
double to
single. For example, when you convert the double-precision number
3.14 to
single, it is rounded by
double(single(3.14) - 3.14) ans = 1.0490e-07
The amount that
3.14 is rounded is less than
eps(single(3.14)) ans = single 2.3842e-07
Avoiding Common Problems with Floating-Point Arithmetic
Almost all operations in MATLAB are performed in double-precision arithmetic conforming to the IEEE standard 754. Because computers only represent numbers to a finite precision (double precision calls for 52 mantissa bits), computations sometimes yield mathematically nonintuitive results. It is important to note that these results are not bugs in MATLAB.
Use the following examples to help you identify these cases:
Example 1 — Round-Off or What You Get Is Not What You Expect
The decimal number
4/3 is not exactly representable as a binary
fraction. For this reason, the following calculation does not give zero, but rather
reveals the quantity
e = 1 - 3*(4/3 - 1) e = 2.2204e-16
0.1 is not exactly representable as a binary number.
Thus, you get the following nonintuitive behavior:
a = 0.0; for i = 1:10 a = a + 0.1; end a == 1 ans = logical 0
Note that the order of operations can matter in the computation:
b = 1e-16 + 1 - 1e-16; c = 1e-16 - 1e-16 + 1; b == c ans = logical 0
There are gaps between floating-point numbers. As the numbers get larger, so do the gaps, as evidenced by:
(2^53 + 1) - 2^53 ans = 0
pi is not really π, it is not surprising that
sin(pi) is not exactly zero:
sin(pi) ans = 1.224646799147353e-16
Example 2 — Catastrophic Cancellation
When subtractions are performed with nearly equal operands, sometimes cancellation can occur unexpectedly. The following is an example of a cancellation caused by swamping (loss of precision that makes the addition insignificant).
sqrt(1e-16 + 1) - 1 ans = 0
Some functions in MATLAB, such as
expm1 and
log1p, may be
used to compensate for the effects of catastrophic cancellation.
Example 3 — Floating-Point Operations and Linear Algebra
Round-off, cancellation, and other traits of floating-point arithmetic combine to
produce startling computations when solving the problems of linear algebra. MATLAB warns that the following matrix
A is ill-conditioned,
and therefore the system
Ax = b may be sensitive to small
A = diag([2 eps]); b = [2; eps]; y = A\b; Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND = 1.110223e-16.
These are only a few of the examples showing how IEEE floating-point arithmetic affects computations in MATLAB. Note that all computations performed in IEEE 754 arithmetic are affected, this includes applications written in C or FORTRAN, as well as MATLAB.
[1] Moler, Cleve. "Floating Points." MATLAB News and Notes. Fall, 1996.
[2] Moler, Cleve. Numerical Computing with MATLAB. Natick, MA: The MathWorks, Inc., 2004.