Combined Preorder and Postorder Traversal Algorithm for the Analysis of Singular Systems by Haar Wavelets

Algorithm 1

Step 1. 𝙸𝚗𝚒𝚝𝚒𝚊𝚕𝚒𝚣𝚎𝐀_ℎ,𝐀_𝐸,𝐓.

Step 2. 𝙾𝚋𝚝𝚊𝚒𝚗𝐕₁⁽²⁾

𝙸𝚗𝚙𝚞𝚝:𝚁𝚎𝚜𝚘𝚕𝚞𝚝𝚒𝚘𝚗𝚜𝚌𝚊𝚕𝚎𝐽

𝑊𝑎𝑣𝑒𝑆𝑜𝑙𝑣𝑒𝑟(𝐽)

{

for(𝑟=2^𝐽−1+1;𝑟<2^𝐽;𝑟=𝑟+2)

{

Solvefor𝐕_𝐽^(𝑟)thesystem(𝐀_ℎ+2𝐀_𝐸)𝐕_𝐽^(𝑟)=𝐓_𝐽^(𝑟)

Update𝐓_𝐽^(𝑟+1)accrodingto𝐓_𝐽^(𝑟+1)=𝐓_𝐽^(𝑟+1)−4𝐀_𝐸𝐕_𝐽^(𝑟)

Solvefor𝐕_𝐽^(𝑟+1)thesystem(𝐀_ℎ+2𝐀_𝐸)𝐕_𝐽^(𝑟+1)=𝐓_𝐽^(𝑟+1)

𝑊𝑎𝑣𝑒𝑇𝑟𝑒𝑒(1,𝐽,𝑟+1)

}

}

𝙸𝚗𝚙𝚞𝚝:𝚛𝚗𝚘𝚒𝚜𝚊𝚗𝚞𝚖𝚋𝚎𝚛𝚘𝚏𝚛𝚎𝚌𝚞𝚛𝚜𝚒𝚟𝚎𝚌𝚊𝚕𝚕.

𝚁𝚎𝚜𝚘𝚕𝚞𝚝𝚒𝚘𝚗𝚜𝚌𝚊𝚕𝚎𝐽

𝑟𝚒𝚜𝚊𝚗𝚘𝚍𝚎𝚗𝚞𝚖𝚋𝚎𝚛.

𝑊𝑎𝑣𝑒𝑇𝑟𝑒𝑒(rno,𝐽,𝑟)

{

if(𝐽−rno≤0)

return

Merge:𝐕^(𝑟/2)_𝐽−rno=𝐕^(𝑟−1)_𝐽−rno+1𝐕^(𝑟)_𝐽−rno+1

𝑟if(2iseven)

𝑟𝑊𝑎𝑣𝑒𝑇𝑟𝑒𝑒(rno+1,𝐽,2);

else

UpdateandSplit:𝐓^(𝑟−1)_𝐽−rno+1𝐓^(𝑟)_𝐽−rno+1=𝐓^(𝑟/2+1)_𝐽−rno−4𝐀_𝐸𝐕^(𝑟/2)_𝐽−rno𝟏_𝑚/2^𝐽−rno

}

Step 3. 𝚂𝚘𝚕𝚟𝚎𝐕₁⁽¹⁾𝚏𝚛𝚘𝚖(4.12).