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Discretization error

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25: 450:. Discretization error would occur even if it were possible to represent the values exactly and use exact arithmetic – it is the error from representing a function by its values at a discrete set of points, not an error in these values. 300: 379: 205: 401: 489: 42: 210: 108: 89: 61: 154: 403:
is a finitely small number, the difference between the first formula and this approximation is known as discretization error.
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Error from taking a finite number of steps in a computation to approximate an infinite process
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variable is represented in the computer by a finite number of evaluations, for example, on a
477: 420: 384: 443: 508: 527: 24: 130: 481: 424: 476:. Other Titles in Applied Mathematics (2 ed.). SIAM. p. 5. 295:{\displaystyle f'(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}} 165:
Discretization error is the principal source of error in methods of
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Discretization error, which arises from finite resolution in the
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are satisfied, otherwise the resulting error is called
387: 374:{\displaystyle f'(x)\approx {\frac {f(x+h)-f(x)}{h}}} 308: 213: 182: 49:. Unsourced material may be challenged and removed. 419:, and results in no loss if the conditions of the 395: 373: 294: 199: 389: 184: 235: 470:Accuracy and Stability of Numerical Algorithms 8: 388: 386: 329: 307: 250: 238: 212: 183: 181: 109:Learn how and when to remove this message 459: 7: 438:, which is finite resolution in the 47:adding citations to reliable sources 415:, the analog of discretization is 14: 176:When we define the derivative of 23: 141:resulting from the fact that a 34:needs additional citations for 362: 356: 347: 335: 323: 317: 283: 277: 268: 256: 242: 228: 222: 194: 188: 1: 434:should not be confused with 550: 173:of computational physics. 467:Higham, Nicholas (2002). 448:floating-point arithmetic 200:{\displaystyle \,\!f(x)} 514:Linear multistep method 482:10.1137/1.9780898718027 397: 375: 296: 201: 171:pseudo-spectral method 58:"Discretization error" 398: 396:{\displaystyle \,\!h} 376: 297: 202: 127:computational physics 385: 306: 211: 180: 135:discretization error 43:improve this article 534:Numerical analysis 519:Quantization error 436:quantization error 393: 371: 292: 249: 197: 167:finite differences 155:computational cost 123:numerical analysis 491:978-0-89871-521-7 442:(values), nor in 413:signal processing 407:Related phenomena 369: 290: 234: 119: 118: 111: 93: 541: 496: 495: 475: 464: 421:sampling theorem 402: 400: 399: 394: 380: 378: 377: 372: 370: 365: 330: 316: 301: 299: 298: 293: 291: 286: 251: 248: 221: 206: 204: 203: 198: 114: 107: 103: 100: 94: 92: 51: 27: 19: 549: 548: 544: 543: 542: 540: 539: 538: 524: 523: 505: 500: 499: 492: 473: 466: 465: 461: 456: 444:round-off error 409: 383: 382: 331: 309: 304: 303: 252: 214: 209: 208: 178: 177: 163: 115: 104: 98: 95: 52: 50: 40: 28: 17: 12: 11: 5: 547: 545: 537: 536: 526: 525: 522: 521: 516: 511: 509:Discretization 504: 501: 498: 497: 490: 458: 457: 455: 452: 408: 405: 392: 368: 364: 361: 358: 355: 352: 349: 346: 343: 340: 337: 334: 328: 325: 322: 319: 315: 312: 289: 285: 282: 279: 276: 273: 270: 267: 264: 261: 258: 255: 247: 244: 241: 237: 233: 230: 227: 224: 220: 217: 196: 193: 190: 187: 162: 159: 117: 116: 31: 29: 22: 15: 13: 10: 9: 6: 4: 3: 2: 546: 535: 532: 531: 529: 520: 517: 515: 512: 510: 507: 506: 502: 493: 487: 483: 479: 472: 471: 463: 460: 453: 451: 449: 446:arising from 445: 441: 437: 433: 428: 426: 422: 418: 414: 406: 404: 390: 366: 359: 353: 350: 344: 341: 338: 332: 326: 320: 313: 310: 287: 280: 274: 271: 265: 262: 259: 253: 245: 239: 231: 225: 218: 215: 191: 185: 174: 172: 168: 160: 158: 156: 152: 148: 144: 140: 136: 132: 128: 124: 113: 110: 102: 99:December 2009 91: 88: 84: 81: 77: 74: 70: 67: 63: 60: –  59: 55: 54:Find sources: 48: 44: 38: 37: 32:This article 30: 26: 21: 20: 469: 462: 439: 431: 429: 410: 175: 164: 134: 120: 105: 96: 86: 79: 72: 65: 53: 41:Please help 36:verification 33: 454:References 147:continuous 131:simulation 69:newspapers 351:− 327:≈ 272:− 243:→ 528:Category 503:See also 425:aliasing 417:sampling 381:, where 314:′ 219:′ 169:and the 161:Examples 143:function 432:domain, 151:lattice 137:is the 83:scholar 488:  129:, and 85:  78:  71:  64:  56:  474:(PDF) 440:range 145:of a 139:error 90:JSTOR 76:books 486:ISBN 62:news 478:doi 411:In 302:or 236:lim 207:as 121:In 45:by 530:: 484:. 427:. 157:. 133:, 125:, 494:. 480:: 391:h 367:h 363:) 360:x 357:( 354:f 348:) 345:h 342:+ 339:x 336:( 333:f 324:) 321:x 318:( 311:f 288:h 284:) 281:x 278:( 275:f 269:) 266:h 263:+ 260:x 257:( 254:f 246:0 240:h 232:= 229:) 226:x 223:( 216:f 195:) 192:x 189:( 186:f 112:) 106:( 101:) 97:( 87:· 80:· 73:· 66:· 39:.

Index


verification
improve this article
adding citations to reliable sources
"Discretization error"
news
newspapers
books
scholar
JSTOR
Learn how and when to remove this message
numerical analysis
computational physics
simulation
error
function
continuous
lattice
computational cost
finite differences
pseudo-spectral method
signal processing
sampling
sampling theorem
aliasing
quantization error
round-off error
floating-point arithmetic
Accuracy and Stability of Numerical Algorithms
doi

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