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Re: Re: Solutions for functions containing jump discontinuities

• To: mathgroup at smc.vnet.net
• Subject: [mg39007] Re: [mg38982] Re: Solutions for functions containing jump discontinuities
• From: Dr Bob <drbob at bigfoot.com>
• Date: Fri, 24 Jan 2003 05:06:09 -0500 (EST)
• References: <b0lvo5$5b2$1@smc.vnet.net> <200301231305.IAA11755@smc.vnet.net> <oprjg2g4k4amtwdy@smtp.cox-internet.com>
• Sender: owner-wri-mathgroup at wolfram.com

Here's a useful plot:

f[x_] := 5(x - 1500 Ceiling[x/1760])
plot1 = Block[{$DisplayFunction = Identity}, Plot[f@x, {x, -2000, 15000}]];
Show[plot1, Graphics@{
        AbsolutePointSize@5,
        Point[{#, f@#}] & /@ Range[0, 9000, 1500]}
    ];

Bobby

On Thu, 23 Jan 2003 15:40:06 -0600, Dr Bob <drbob at bigfoot.com> wrote:

> You missed two roots (0 and 1500).  Here's the case Mod[x, 1760] > 0:
>
> f[x_] := 5(x - 1500 Ceiling[x/1760])
> xRule = x -> 1760 k + y;
> Simplify[f@x /. xRule, {0 < y < 1760, k â?? Integers}]
> % /. Ceiling[a_] -> 1
> yRule = First@Solve[% == 0, y]
> Flatten[k /. Solve[y == # /. yRule, k] & /@ {1, 1759}]
> kValues = Range[Ceiling@Min@%, Floor@Max@%]
> yValues = y /. (yRule /. List /@ Thread[k -> kValues])
> 1760kValues + yValues
> f /@ %
>
> and here's the case Mod[x, 1760] == 0:
>
> xRule = x -> 1760 k + y;
> Simplify[5(x - 1500 Ceiling[x/1760]) //. {xRule, y -> 0}, {k â?? Integers}]
> yRule = First@Solve[% == 0, k]
> 1760k /. %
> f@%
>
> Bobby
>
> On Thu, 23 Jan 2003 08:05:15 -0500 (EST), Orestis Vantzos 
> <atelesforos at hotmail.com> wrote:
>
>> Your function simplifies to:
>> f[x_]:=5(x - 1500 Ceiling[x/1760])
>>
>> Now assume that [First Case] x==1760 k + y , 0<y<=1760 and k Integer
>> Then Ceiling[x/1760]== Ceiling[k + y/1760]== k+1
>> so that f[x]==5 (-1500 + 260 k + y)
>>
>> If f[x]==0 then 5 (-1500 + 260 k + y)==0 and we solve for y:
>> y== 1500-260 k
>>
>> 0<y<=1760 =>
>> 0< 1500-260k <=1760
>> -1500<-260 k <= 260
>> 5.77 > k >= 1
>>
>> So k ranges from 1 to 5 and since x==1500(k+1) the roots are:
>> Table[1500(k + 1),{k,1,5}]
>> {3000, 4500, 6000, 7500, 9000}
>>
>> Orestis Vantzos
>>
>> newspostings at burkert.de (Burkert, Philipp) wrote in message 
>> news:<b0lvo5$5b2$1 at smc.vnet.net>...
>>> Hi folks,
>>>
>>> we are searching all solutions where the function f results null.
>>>
>>> f[x_] := -7500 * Ceiling[(0.5 * x) / 880] + (5 * x)
>>> Solve[{f[x] == 0}, x]
>>>
>>> As f contains jump discontinuities, we recieved the following error:
>>>
>>> InverseFunction::"ifun": "Inverse functions are being used. Values may 
>>> be \
>>> lost for multivalued inverses."
>>>
>>> Solve::"tdep": "The equations appear to involve the variables to be 
>>> solved \
>>> for in an essentially non-algebraic way."
>>>
>>> We would be pleased if anybody could help us.
>>>
>>> Regards,
>>> Philipp Burkert
>>> Carsten Siegmund
>>
>>
>
>
>



-- 
majort at cox-internet.com
Bobby R. Treat

• References:
  • Re: Solutions for functions containing jump discontinuities
    • From: atelesforos@hotmail.com (Orestis Vantzos)